ON JONES KNOT INVARIANTS AND VASSILIEV INVARIANTS By Jun Zhu B. Sc. (Mathematics) Suzhou University , 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES THE DEPARTMENT OF MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1995 © Jun Zhu, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) s Department of The University of British Columbia Vancouver, Canada Date DE-6 (2/88) A-pr. ], Abstract The main objective of this thesis is to study invariants of knots and links. First, a minimal system is associated to each link diagram D. This minimal system is an “an tichain” in the partially ordered set of all possible functions from the crossings of D to {—1, 1}. Such a system is then shown to be an effective tool for determining the highest degree as well as the span of the Kauffman bracket (D). When applied to a diagram D with n crossings which is “dealternator connected” and “rn-alternating”, the upper bound spanKD) 4(n—m) is obtained. This is a best possible result providing a negative answer to the conjecture posed by C.Adams et al. Next, some examples are presented to show that a semi-alternating diagram need not be minimal. This disproves a conjecture posed by K.Murasugi. For knots and links, if J(t) is a so-called generalized Jones invariant, then J()(i), the n-th derivative of J(t) evaluated at t = 1, is a Vassiliev invariant. On the other hand, while the coefficients of the classical Conway polynomial are Vassiliev invariants, the coefficients of the Jones polynomial are now shown not to be Vassiliev invariants. An interesting property of Vassiliev invariants is then given to prove many known results in a uniform manner. It is known that all the Vassiliev invariants with order less than or equal to n form a vector space called the n-th Vassiliev space. It turns out that two lens spaces have the same Vassiliev spaces if and only if their fundamental groups are isomorphic. Consequently, Vassiliev spaces do not distinguish manifolds. ii Table of Contents Abstract ii List of Figures v Acknowledgements vi 1 Introduction 1 2 Kauffman bracket 5 2.1 Introduction 5 2.2 Minimal System 6 2.3 Degree of Kauffman bracket 10 2.4 The Kauffman bracket and binomial coefficients 16 2.5 rn-almost alternating diagrams 20 2.6 Semi-alternating diagrams 25 3 4 Jones knot invariants and Vassiliev invariants 30 3.1 Introduction 30 3.2 Relationship between quantum group invariants and Vassiliev invariants 31 3.3 Some known invariants which are not Vassiliev invariants 34 3.4 Numerical invariants which are approximable by Vassiliev invariants 3.5 Knot connected sum in finite type spaces 44 3.6 Some properties of Vassiliev invariants 48 . 40 Finite type invariants for lens spaces 51 4.1 Introduction 51 4.2 An operation on finite type spaces 52 111 4.3 Similarity classes 57 4.4 Finite type spaces for lens spaces 62 Bibliography 66 iv List of Figures 2.1 Two different smoothings 2.2 A non-adequate knot diagram 2.3 A diagram for the unknot 16 2.4 Link 97 19 2.5 A counterexample 23 2.6 An alternating diagram for the counterexample 24 2.7 Signed crossings 26 2.8 A link diagram D and related graphs G, F 0 (G) and F(G) 27 2.9 An equivalent diagram 28 3.10 Related knot diagrams 31 3.11 Torus knot of type (2, 2n + 1) 34 3.12 A singular torus knot 35 3.13 A singular torus knot with 2n double points 36 6 . 3.14 A singular torus knot with 2n + 1 double points . . . . 10 39 4.15 Connected sum of two knots in solid torus 55 4.16 Arc winding numbers 60 v Acknowledgement It is a great pleasure for me to thank my supervisors, Dr. Dale Rolfsen and Dr Kee Lam, who spent lots of valuable time to guide me through my research and to read a draft of this thesis. Special thanks also go to professors Joan Birman and Xiao-Song Lin of Columbia University for their useful comments. Finally, I would like to thank my wife Yunfang for her devotion and support. vi Chapter 1 Introduction A knot (link) K is an embedding of an (several copies of) oriented circle 5’ in 3-space 1R 3 or the 3-dimensional sphere S . The knot type of K is the topological type of the pair 3 , K), under homeomorphisms which preserve orientations on both S 3 (S 3 and K. When we say “two different knots”, we usually mean “two different knot types”. The theory of classical knots in 3-dimensional space iii has played a very important, if not crucial, role in the theory of 3-dimensional manifolds. The most fundamental problem in knot theory is the classification problem of knot types. The first systematic contribution to this problem is by P.G.Tait and his coworkers [67] a century ago. They compiled a table of knots with relatively few crossings. Now knots have, in theory, been classified by Haken [24] and Hemion [25]. But the classification is by means of an al gorithm that is too complex to use in practice. On the other hand, the complementary space of a knot determines the knot type [79] and [22]. However, the topological equiv alence classes of the complementary spaces of knots is in general not easy to determine. Thus one is led to seek simple invariants for knots which classify large classes of specific knots. The Alexander polynomial [3], /jc(t), a Laurent polynomial in variable t, is one such knot invariant which was and is still a most useful one. Let us denote by K, K_ and K 0 the knots which are defined by diagrams that are identical outside a neighborhood of a particular double point, and differing in the manner indicated in Figure 3.10. Then the Alexander polynomial is characterized by the following crossing change 1 Chapter 1. Introduction 2 formula — together with zXu = = (— 1, where U is the unknot. Modifying the formula to J(K+) 1 r together with J(U) = — tJ(K_) = (v’ — 1, one can obtain another very different polynomial invariant, nowadays called the Jones polynomial. This is a remarkable discovery by V. Jones [30]. Although there are infinitely many knots sharing the same Jones polynomial, see [33], whether the Jones polynomial can distinguish nontrivial knots from the unknot is still an important open problem. See [39] and [64]. It is well-known that the Jones polynomial is retrievable from the Kauffman bracket. Now the problem can be restated as to whether there is a knotted link diagram whose Kauffman bracket is a monomial. Based on knowledge on the bracket, Kauffman [36], K. Murasugi [56] and M. Thistlethwaite [69] settled some hundred-year-old conjectures of P.O. Tait which imply that the above problem on Kauffman bracket can be answered affirmatively within the family of reduced alternating link diagrams. Later, Lickorish and Thistlethwaite [50] generalized this to the wider family of adequate link diagrams. In Chapter 2 of this thesis, we generalize this further to subadequate link diagrams. Based on our techniques, we are able to solve a conjecture of C. Adams and his coworkers [1] and a conjecture of K.Murasugi [58]. Motivated by the discovery of the Jones polynomial, more knot polynomials were subsequently found. Notably, the HOMFLY polynomial and the Kauffman polynomial. A unified setting for finding such knot invariants is available now, see [32] and [74]. All such invariants are called generalized Jones invariants. Since generalized Jones invariants do not distinguish a knot from its reverse (a knot obtained from the original knot by Chapter 1. Introduction 3 reversing orientation) and since a knot may not be equivalent to its reverse [73], the generalized Jones invariants are not enough to classify knots. Around 1990, Vassiliev [75], using an approach pioneered by Arnold, introduced a set of numerical knot invariants via a totally different method. Instead of studying one single knot at a time, he studies the space of all knots and constructed a sequence of subspaces in the 0-th cohomology group of the space of all knots. These spaces have become known as Vassiliev spaces and the elements in Vassiliev spaces are called Vassiliev invariants. Although the generalized Jones invariants and the Vassiliev invariants come from different viewpoints, they are closely related. For example, Bar-Natan [7] proved that every coefficient of the Alexander Conway polynomial is a Vassiliev invariant. Actually, a deep relationship between generalized Jones invariants and Vassiliev invariants has been found in [13] and [44]. There the authors show that, after a suitable change of variables, each coefficient in the Taylor series expansion of a generalized Jones invariant is a Vassiliev invariant. The first result in Chapter 3 is that every coefficient in the Taylor series expansion at 1 of a generalized Jones invariant is a Vassiliev invariant. In other words, the n-th derivative of a generalized Jones invariant evaluated at 1 is a Vassiliev invariant. In contrast with the Alexander Conway polynomial, we show that every coefficient of the Jones polynomial is not a Vassiliev invariant. Hence the classical Alexander Conway polynomial and the Jones polynomial are, in this sense, quite different. Whether every numerical invariant can be approximated by Vassiliev invariants is an important open problem, see [13]. We prove in Chapter 3 that the space of numerical invariants which can be approximated by Vassiliev invariants is a dual space of some vector space. Finally, we give an interesting property of Vassiliev invariants from which we can retrieve, in a uniform manner, many known results. In Chapter 4, we study Vassiliev spaces for 3-manifolds, in particular for the lens Chapter 1. Introduction 4 spaces. We show that two lens spaces have isomorphic Vassiliev spaces if and only if their fundamental groups are isomorphic. Consequently, we see that Vassiliev spaces do not distinguish manifolds. Chapter 2 Kauffman bracket 2.1 Introduction Since V. Jones invented his knot invariants [30], many applications have been found. One of the most important applications is a solution to a hundred-year-old conjecture of P. G. Tait. See [36], [56] and [69]. The key method Kauffman used to settle this conjecture is to determine the span of the Kauffman bracket for reduced alternating link diagrams; i.e., to determine the highest and the lowest degree of the Kauffman bracket for such link diagrams. W. B. R. Lickorish and M. B. Thistlethwaite generalized this method to adequate link diagrams [50]. In this chapter, we generalize the method to a larger class of link diagrams by associating a minimal system to each link diagram (for the definition see the following section), which is closely related to the highest or the lowest degree of the Kauffman bracket. Our approach is easy to apply and allows us to generalize some results of [36] and [50]. It is also interesting to note that our method can be used to prove some identities on binomial coefficients via Kauffman bracket. See section 2.4. In their work [1], the authors defined rn-almost alternating projections of n crossings for a link. (We may call such a projection an rn-almost alternating diagram). They proved that for a dealternator reduced and dealternator connected rn-alternating diagram D, the span of its Kauffman bracket is less than or equal to 4(n the span is less than or equal to 4(n — m — — m — 2), and conjectured that 1) if the condition “dealternator reduced” is removed. As an application of our methods, we show that the conjecture is true if one 5 Chapter 2. Kauffman bracket 6 /\ /\ A-smoothing A’-smoothing Figure 2.1: Two different smoothings replaces 4(n — m — 1) by 4(n — m). An example is then provided to show that this upper bound is best possible. The conjecture is hence completely solved. K.Murasugi conjectured in [58] that every semi-alternating link diagram is minimal, namely that it realizes the minimal crossing number for the link. We construct a sequence of semi-alternating link diagrams which are not minimal. Hence Murasugi’s conjecture is false. In [58] K. Murasugi also raised the question whether a semi-alternating link diagram is adequate. Our semi-alternating link diagrams turn out to be not adequate. The answer is thus negative as well. In what follows, by a diagram we always mean a link diagram which is connected as a graph. 2.2 Minimal System let D be a diagram for some link, with crossings c ,c 1 ,.. 2 {cI1 i n} —+ {1, —1}. according to an A-smoothing if .s(c) 2.1. , c. A state for D is a function Let sD be the diagram with its crossings nullified = 1, and an A -smoothing if .s(c) 1 = —1, see figure Chapter 2. Kauffman bracket 7 We abbreviate s(c) by s(i), and denote the number of components of sD by Let S {ss is a state for D}. Then we can impose a natural partial order namely, s <s’ if and only if s(i) s’(i), for every i with 1 IsDI. on 5, i <n. This turns S into a poset, or partially ordered set. Denote Sq then S = = {s E Ss(i) n = — 2q},O q n, {1}, where 1 is the constant map taking value 1, and similarly, Si-, Lemma 1 Let s, s’ € S with s {—1}. = s’, then s’(i)+2s’D and equality holds if and only if sD 2r Proof. Let s E Sq. Since s Zs(i) = si(i) + 2 for 0 Note that sDI = s(i) = — — r, where r satisfies s’(i). r = 5 s’ such that r. We then have 5 p E5 q+p for every q. p D + 1 and s(i) 1 s = = 2+s (i), hence 1 s÷ ( 1 i) + 2 + 2(Is+iD + 1) s ( 1 i) + 1 2Is+ D I+2±2 and equality holds if and only if si > s’, there are s s(i) + 2IsDI = s’D = IsDI = s ( 1 i) + 2s+iDI D 1 s — 1. It is then easy to see s(i)+2IsD > s’(i)+2Is’D and equality holds if and only if jsD 0 = Is’D — r, where r satisfies 2r = s(i) — s’(i). Chapter 2. Kauffman bracket Remark 1 If sD 8 D + 1, then 1 s = s(i) + 2sD —4 = (i) + 1 s Recall that the Kauffman bracket for diagram D is defined as follows: (D) = (Ds) sES where (Djs) From the definition of than or equal to equal to s(i) — = 2 AE5(i)(_A_ — )IsDI_1. 2 A (Dis), it is easy to see that the highest degree of (Ds) is less s(i) + 2sDI — 2 and that the lowest degree of Ds) is greater than or 2sD + 2. Therefore, by Lemma 1, we have the following well known result: Corollary 1 The highest degree of (D) does not exceed n + 21D degree of KD) is at least—n — 2 — 0, there are unique elements s (a) {s,. Si . , sk} 2 and the lowest 1D +2. Proposition 1 Let D be a diagram with crossings c , 1 q — ,5j , c,-. Then for every integer ES (up to permutation) such that is an antichain in the poset 5, i.e. for every pair Si, 5j with i j, Si, (b) For every r 8 E 5r() {Si, . , 1(i) + 21D + 2ISrDI — 4q = n + 21D — 4q. and (c) {s,. is minimal, i.e. if {s’,... every s, there exists an r 5 E {si,. . ,4} C S satisfies (a) and (b), then for such that s, Chapter 2. Kauffman bracket 9 Proof. Let = Clearly, 1 Let no s e B {Si,• e B , .Sk} with s {s E S s(i) + 2sD n + 21D {si,. . , 4q}. 0. be the set of all minimal elements in B. i.e. for every r such that s < r• It is clear that {s,. (b). To show (c), it suffices to show that for any s r — s,} such that .s r. . . , r 3 there is sj} satisfies (a) and $ satisfying (b) there exists an But this clearly follows from the definition of Bc, hence the proposition. Definition 1 The antichain {s,. . (b) and (c) in Proposition 1 is ,s,} satisfying (a), called the q-th minimal system of the diagram D. The O-th minimal system is called the minimal system. Remark 2 is, in fact, equal to 1 3j, where U {s j = $s the closure of the q-th minimal system. By Lemma 1, B C smallest integer such that B S, then 11D1 + n 2q If q is the smallest integer such that BD) — = = S, where I — s}. Let us call for q 1DI, and — q’. If q is the 1DI n+1 — q. is the mirror image of D, then q+=n and 1D<n+1—q. Remark 3 Recall that the diagram D is +adequate if 1DI > sDI for every s E Si and is adequate if it is +adequate and the mirror image is also +adequate. Hence for adequate or +adequate diagrams, their minimal systems are always {l}. Example 1. Let D be a diagram of the knot 101.5.6. We draw a diagram of 1D by nullifying its crossings with respect to state 1. See figure 2.2. From the diagram of 1D, one can easily find its minimal system 125 3345, 367, 81267} S {, Chapter 2. Kauffman bracket 10 Knot 10156 1D Figure 2.2: A non-adequate knot diagram where ijk 5 is a state taking value —1 only at {i,j,. , k}. By the above remark, D is not +adequate. Let us finish this section with the following lemma which can be easily checked. Actually it is a special case of Proposition 2 in section 2.4. Lemma 2 0 r (_1)r() 0 = n = k=O 2.3 ( r <n Degree of Kauffman bracket Let us denote by DegD) the highest degree of a Kauffman bracket, hereafter simply referred to as the degree of the Kauffman bracket for D. If D denotes the mirror image of the diagram D, then the lowest degree of (D) is equal to —Deg(D). Hence, to know the lowest degree of (D), we only need to know how to determine the degree of Kauffman bracket. Chapter 2. Kauffman bracket Lemma 3 Suppose = {Si, 2,••• {s 6 Ss > s}. Let B 11 , sk} is an antichain in S satisfying fl 1 that is, B = = {1}, where B° in the notation of Remark S above. Then (_1)iBnSjI 0, = where X denotes the number of elements in the set X. Proof. We prove the lemma by induction on k. For k implies s 1 i fl Si I = 1, hence we may assume s 6 i=o Now assume k > 1, let B 1 = function given by in 1 and s > s {Si A 2, 1, note that the assumption for some r > 0, and since lB fl S = (), it follows from Lemma 2 that nS s r 8 = 1A s i r 8 A s(i) for some 33, , A ., = 2 B = (j (_1) =0. J i=o and B’ = B 1 A sj to be the 1 fl B . Define s 2 max{si(i),s(i)}, 1 n. Notice that for every s e B’, i r, 5 hence the set of all minimal elements of B’ is contained sk}. This set is clearly an antichain satisfying the conditions in the present lemma. Since the number of minimal elements in B’ is less than k, by induction assumption we have l 3 (1)’lB’nS = 0, and with the same reason, (_1)ilBi n S = 0 and (_1)iIB n 2 Sl =0. Hence n Sl = (_1)iIBi n Sl + (—1)IB 2 n S1 — (_1)uIB n Sl = 0. Chapter 2. Kauffman bracket 12 0 Corollary 2 For an antichain s, we have (—1)IBflSl 0 if and only if s = 1. We may generalize Lemma 3 to the following Lemma 4 Suppose with 0 < r Fr_i} and < , 4},. , {s,. , s} are antichains in S. If for every r p, there exists a partition {Fr_i, Gr_i} of {1, 2,•• , kr_i} such that {s’li E {sli E Gr_i} satisfy the conditions in Lemma Sand set of all minimal elements in (UjEFr_ij)fl(UjEGr_ij) ‘r} is the and iffurther{s,... , s} satisfies the conditions in Lemma 3, then (1)ilB n SI = 0 where B = Proof. The case p = 0 is proved in the above lemma. Assuming the lemma true up to <p—i, we let B 2 1 = UEFO.i and B {sç,. , = UEGOi so that B 1 flB 2 = and {s,... , s,,} satisfy the conditions in the present lemma. By induction assumption we have 2 n S1 (—1)lBi n B = 0, and by Lemma 3 (_1)ilB nSl = 0, i = 1,2. Hence n s1 = (_1)uIBi n sl + (—1)IB 2 n Sl — 2 fl (—1)lBi fl B SI =0. Chapter 2. Kauffman bracket 13 Theorem 1 Suppose a link diagram D with n crossings has minimal system playing the role of {$,. , 4} in Lemma . {Si, 2, , sk} Then Deg(D) <n + 2I1D — 2. by B as before. Then by Lemma 1 and the definition of Proof. Denote >Z Kauffman bracket, it is not difficult to see that the term of degree n + 2I1D — 2 for (D) is (_l)ISDI_1A8(i)+2ISDI_2 = sEB (_l)18D1_1AEs(i)+218D1_2 j=OsEBnS (_l)I1DI+i_1AE1()+2I1DI_2 = j0 sEBflS 3 (_l)11D1+i_1Afl+211D1_2 = j0 sEBflS 3 = Therefore, Deg(D) < n + 21D — {(_1)1lDI_l(_1)iB n 2. D Remark 4 The proof of the theorem indicates that the term of degree n+21D —2—4q is related only to B, B , 1 term of degree n + 21DI .. — , For example, if we know B, B , we can determine the 1 6. Now we try to decide when Deg(D) is equal to n + 21D Lemma 5 Suppose {$,. . , $},... , {s,. . , s} . — 2. are antichains in S satisfying all the conditions in Lemma 4 except that the last condition is replaced by {sç,. Then nS where B = = (-i) . , s} = {1}. Chapter 2. Kauffman bracket 14 Proof. Just mimic the proof of Lemma 4, noting that oniy the last term in the sum is nonzero and equal to (_1)1’. We leave the details to the reader. Definition 2 A link diagram D with a minimal system {si,. if there are antichains {s,. . ,{5,. . . . . . , Sk} ,s} with {s,. 0 is called +subadequate ,s} = such that these antichains satisfy the conditions in Lemma 5. D is called subadequate if both D and the mirror image Ti of D are +subadequate. Remark 5 Clearly + adequate diagrams are +subadequate diagrams. Hence adequate diagrams are subadequate. Example 1 in section 2.2 is not +adequate but is +subadequate. In fact, and 2 B {5367, 51267} U = {512} and {53} form a partition for the minimal system. Let B 1 Then {512, 53} is an antichain with B 1 fl B 2 form a partition satisfying fl Moreover, by the following theorem, Deg(D) = = {3125,5345} U = U = . and Clearly {1}. It is therefore +subadequate. 10 + 4 — 2 = 12. It is also easy to find a subadequate diagram which is not adequate. For example, a diagram with two crossings for the trivial link of two components is subadequate, but not adequate. Theorem 2 If D is a +subadequate diagram, then Deg(D) = n + 21D —2. Proof. As in the proof of Theorem 1, the term of degree n + 21D (_l)I8DI_1AES(i)+2ISDI_2 — 2 for (D) is (l)(SDI_1AES(i)+213D1_1 = j=O seBflS, séB = (_l)I1DI_1((_l)iIB n Sj)A2IDI_2. Chapter 2. Kauffman bracket 15 By Lemma 5, the number inside the parenthesis is nonzero, therefore, by Corollary 1, Deg(D) = n + 21D Corollary 3 Suppose D have minimal system {s,. DegD) = n + 21D o 2. — . Sk} , — (k < 2). Then 2 if and only if = {1} The proof of this corollary is trivial. Using subadequate diagrams, we can generalize many results in [361 and [50]. We will not go over these one by one, but only mention the following corollary. Recall that Deg(D) + Deg(D), then we have .span(D) Corollary 4 If D is subadequate, then span(D) 2n + 2(1D + particular, if n> 1, then span(D) > 0 and the Jones polynomial VD(t) Lemma 6 Let {5i,• with 1 , 2 of{1,...,k} such that fl{ i e F} {1}forl p = {1}. Then (—1)BnSI= 1—1 = Proof. Denote B 1D) — = U{i E F}. By the assumption we have In 4. 1. ,. 1 sk} be an antichain in S. Suppose there is a partition {F (U{i E F}) fl (U{i E Fq}) where B — . . , Fi} 1, and that forp q, Chapter 2. Kauffman bracket 16 n crossings Figure 2.3: A diagram for the unknot for j > 0, hence = BflsoI+(—1)BnsI = j=1 p=l n = j=O in = p=l j=O = 1—i. The last equality follows from Lemma 4. 0 By the above lemma, we can generalize Theorem 2 to some other cases. We leave this to the reader. 2.4 The Kauffman bracket and binomial coefficients In this section we prove some identities on binomial coefficients via Kauffman bracket. In fact we only use a very simple diagram; the same idea can be applied to other diagrams. We do not know whether one can find some new or really interesting binomial coefficient identities in this way. Proposition 2 Z (—i)() 0 () = 0 for 0 k < n, where () = 0 if i <k. Chapter 2. Kauffman bracket 17 Proof. Let D be the diagram shown in figure 2.3. Then the minimal system is clearly {—1} and 1D = 1, hence B = S and sD = 1 +j for every .s e S,. By the definition of The Kauffman bracket we have (D) = (A — A 2 A ) 2 i0 sES, = A (—A — A 2 )S 2 = (_1)i[()An + ()An-4 + ... + ()An-4k + ... + -r() (+ +[ () (;)n () () (i () () (;)() (j- () (j(j (:) (;)n + +(1)u[() (jAn + +(1)n[() (:)Afl ()Am-4i1 + +... (jAn-4 + ... + + (:)A4i + Therefore the coefficient of A’ ’’, summed along the k-th column, is 4 () (j = E(1)i () (j. On the other hand, (D) = A , hence for k < n we have 3 () (;) Remark 6 Some special cases, for example, known. = 0. (_i)i() 0 D = 0 when k = 0, are well Chapter 2. Kauffman bracket 18 As an application, we prove the following theorem which allows us to compute the highest degree of Kauffman bracket in some non +subadequate cases. See Example 2 below. Before stating the theorem, we need the fact that if then fl 1 = for some s E S. For instance, sj fl . = {Sj,. Si , .s} is an antichain in S, A .s. Theorem 3 Suppose D is a link diagram with minimal system {s,... for some s , Sk}. If = Sr then the highest degree of 2 AE)(A — )I°I’ 2 A sEB is less than n + 2I1D —2— 4r. Here B = Proof. The case k = 1 follows from the above proposition. Mimic the proofs of Lemma 3 and Theorem 1 for the general case. 0 Example 2. It is not difficult to see that the diagram D in Figure 2.4 has minimal system {S12}, hence D is not +subadequate, and that its first minimal system is 1234 1256, S3456}. { 127, 128, 129, S Choose a partition as follows {{S127, 128, 129, 51256}, {31234, 53456}}, 1 fl B B 2 where B 1 = Since 12 127 fl U 8128 56 = U 129 U 1256 = 12 and B 2 U then 56 = 81234 U 83456. {1} the conditions in Lemma 5 are satisfied. Therefore, we have (—1)jB’nS= = 1, Chapter 2. Kauffman bracket 19 64)5 Link 97 1D Figure 2.4: Link where B’ = 8127 U 8128 U 129 U 1234 U 1256 U 3456• Hence the highest degree of 2 AES(i)(_A — AE8()(_A2 )I + 2 A I1 13 sEB’—B — )I9D1_ 2 A 3 sEB is n + 21Dj —6 Denote 12 by B. Applying the above theorem to conclude that the highest degree of of SEB 2 A()(—A — )I’I 2 A 3 SEB 2 AE5(i)(_A SEB — 2 A()(—A )I 2 A 11 are both less than n + 21D — 2 — — 2 AE5(z)(_A_ )15D1_ + 2 A 1 sEB sEW—B which is n + 21DI — 6 — we and the highest degree — 8 = n + 211D1 Hence, the highest degree of (D) is equal to the highest degree of 2 AE5()(_A_ )15D1_l, 2 A )15D1_ 2 A 3 — 10. Chapter 2. Kauffman bracket 2.5 20 rn-almost alternating diagrams First of all, let us recall some concepts from [1]. A link diagram is said to be alternating if the pattern of over and undercrossings alternates as one traverses a component. A link diagram is called rn-almost alternating if m is the smallest integer such that m crossing changes produce an alternating diagram. Hence a 0-almost alternating diagram is an alternating diagram, and we call 1-almost alternating diagram simply almost alternating. Let D be an rn-almost alternating diagram with crossings c ,c 1 ,• 2 ,c,-. Then we may assume that changing the first m crossings of D produces an alternating diagram. Let D’ be a diagram obtained from D by nullifying the first m crossings. Clearly, we 2 such diagrams corresponding to different smoothings. A crossing of D is called have m nugatory if some nullyfying of the crossing disconnects D. D is reduced if D has no nugatory crossings. D is called “dealternator reduced” if every D’ is reduced. D is called “dealternator connected” if every D’ is connected. A loop ir is called a “dealternator severing path” if ir intersects D only at some of the first m crossings. It is clear that D is dealternator connected if and only if D has no dealternator severing paths. Lemma 7 Let D be a diagram with minimal system {s, S , 2 Degc(D) n — 2m + 2sDI — ,Sk}. Ifs E B Ft Sm then 2 where B = Proof. By Corollary 1, we have Deg(D) n + 21DI — 2. It follows from the definition of B that 1DI = sD see that the lemma is true. — m. It is then not difficult to o Chapter 2. Kauffman bracket 21 Let i be the mirror image of D. By the obvious one-one correspondence between the crossings of D and the crossings of D, We can think of a state s of D as being also a state of D. So we can denote the set of states for tY by S as well. Denote by B and the closure of the minimal systems for D and 7 respectively. Recall that the span of (D) is the highest degree of (D) minus the lowest degree of (D), that is, Deg(D) + DegD). We have the following Corollary 5 Suppose .s Sm is a state for a diagram D. If s e B fl B, then span(D) 4(77 — m). Proof. By the above lemma we have span(D) 2(n — 2m) + 2{sD + IsDI} — 4. Following Kauffman [36] we have sD+sD<n+2. Hence the inequality. D Proposition 3 Let D be a link diagram and s be a state of D in Sm. If each diagram D’ obtained from D by nullifying the crossings at which s takes value —1 is +adequate, then the closure of the minimal system associated to D is contained in the closure of s; that is, B C .. Proof. Let s’ , then s’ A s . and there exists an s” with s’ < s” < s’ A s such that s” and s’ A s take different values only at one crossing, say c. Notice that ifs” is not in B then neither is s’. Hence we may assume s’ s” without loss of generality. We claim s’D < s’ A sD so that s’ is not contained in B by the definition of B. To Chapter 2. Kauffman bracket 22 justify our claim, we may regard s’ and s’ A s as states for D’, where D’ is a diagram obtained from D by nullifying all crossings at which s takes value —1 in A’-direction if s’ A s takes value +1. Clearly, as states for D’, s’ assumption, D’ is adequate. Hence s’Dl = Is’D’l (D’) and s’ A s 1 S < 1D9 = = 1 in S(D’). By sDj. D Remark 7 The above proposition is also true if we replace s by an antichain in S. Corollary 6 If D is a dealternator reduced rn-almost alternating diagram, then the clo sure of its minimal system is contained in if and only ifi , where s is a state of D satisfying s(i) = —1 m. Proof. This follows from proposition 3 and the fact that each D’ is a reduced alter nating diagram, hence each D’ is +adequate. In what follows we always assume D is reduced, namely, has no nugatory crossings. Proposition 4 Suppose D is a dealternator connected and rn-almost alternating diagram and .s is a state of D satisfying s(i) = —1 if and only if i rn. Then B contains . Proof. The proof is essentially contained in the proof of Lemma 4.3 in [1], we do not repeat it. D By Corollary 6 and Proposition 4 we have Corollary 7 If D is a dealternator reduced and dealternator connected rn-almost alter nating diagram, then D has minimal system {s}, where s is defined as in proposition 4. Proposition 5 If D is dealternator reduced, or dealternator connected, or rn-almost alternating, then so is . Moreover, if changing the first rn crossings in D produces an alternating diagram, then the same is true for D. Chapter 2. Kauffman bracket 23 8( D 1D -1D Figure 2.5: A coiinterexample Theorem 4 If D is a dealternator connected rn-almost alternating diagram, then span(D) 4(n-m). Proof. By Proposition 4, B contains s e B fl . By Corollary 5, span(D) Note that Deg(D) < n + 2I1D — . 4(n By Proposition 5, B also contains — . Hence rn). 2 for rn 0 n + 211D1 1 implies that Deg(D) — 6 by the definition of Kauffman bracket. If D is a dealternator reduced and dealternator connected rn-almost alternating diagram, then, by Corollary 7 and Theorem 1, Deg(D) < n + 21DI — n + 21DI —2 2. — Thus Deg(D) n + 2I1D — 2 — 4. Similarly we have Deg(D) 4. Therefore span(D) 2(n — 2rn) + 2{sD + s} and we have proved the following theorem — 4 — 8 4(n — m — 2) Chapter 2. Kauffman bracket 24 Figure 2.6: An alternating diagram for the counterexample Theorem 5 (Theorem 4 in [1]). If D is a dealternator reduced and dealternator con nected rn-almost alternating diagram, then span(D) 4(n — rn 2). — In the paper [1], the authors conjectured that removal of the condition “dealternator reduced” in the above theorem would lead to span(D) 4(n — rn — 1). It turns out that Theorem 4 is the optimal result (see example 3). Example 3. Let D be the diagram in Figure 1.5. Clearly, D is a dealternator connected 2-almost alternating diagram. From diagrams 1D and —1D, we see that D has minimal system {512, 845} and state taking value —1 only at i and has minimal system j. {512, 33}, where 5j denotes the It is then easy to see that D is subadequate. By Theorem 2 we have span(D) which is exactly 4(n — = 2n + 2(I1D + ) —4 32 4(10 — 2). m). Remark 8 We can compute the span by isotoping the diagram D into an alternating diagram, (Figure 2.6), because the span of D and that of the alternating diagram are the Chapter 2. Kauffman bracket 25 same. 2.6 Semi-alternating diagrams In this section, we will resolve some problems in [58]. To this end, we first fix notations and definitions. We refer to [58] for more details. Let G be a graph. Let V(G) and E(G) be the (finite) sets of vertices and edges of C, respectively. A graph C is said to be signed if either +1 or —1, called sign, is assigned to each edge. More precisely, C is a signed graph if G is a graph equipped with a sign function fG : E(G) —+ {1, —1}. We could call an edge e positive if fG(e) = 1 and negative otherwise. A graph C is said to be separable if there are two subgraphs H and K such that C = H U K and H fl K = }, where both H and K have at least one edge and v 0 {v 0 is a vertex. Let D be a link diagram of a link L. D divides R 2 into a finite number of domains. We shade the domains in checkerboard fashion with the infinite domain shaded. Take a point v from each unshaded domain R. These points form a set of vertices of a graph G. For any two unshaded domains R and 1?3 meeting at a crossing c of D, we assign an edge e to C to join vertices v and vj and sign it according to whether the crossing c is positive or negative. See Figure 2.7. The signed plane graph C obtained above is called the graph of the link diagram D. Conversely, given a signed graph C, one can construct uniquely the link diagram D of a link having G as its signed graph. Let C be a signed plane graph. For a vertex v of G, let 1 e2 , e , • e be an enu meration of all edges emerging from v in the counter-clockwise order. Set (v) = Chapter 2. Kauffman bracket 26 //\ sign sign=—l +1 Figure 2.7: Signed crossings =i sign(ej)sign(eji), e 1 nation index at v. -y(G) = . Then y(v) 1 e = = {n + (v)} 0 is called the alter minvy(v) is called the alternation index of G, where the minimum is taken over all vertices of G. Now we construct two graphs P (G) and F(G) 0 from C as follows. First we subdivide G by adding one vertex w to each edge e of G so that e, is divided into two edges e, e’. The resulting graph is denoted by G’. sign(e) F(v) = = sign(e7) sign(ej), C’ becomes a signed plane graph. = ,i 1 (_l)i+ , ,••• 2 e2, = . , . e2k 1 , 3 e , , 3 _1, e 3 e3,fl 2 , 4 e , ,, 4 e , , 4 e , e2k,1,• , k_1, e2k, 2 e2k,fl , 1 , 2 ei,,, e e2k,k, , , W2k+1,1 = = e, e ,••., , 2 , and their ends. F(v), ,e 2 2 , 3 ,e 1 _ 2 e2, and their ends. Here, ëj(l joining two vertices wi,, 1 and wii,i, where Then 1’ (v) is defined 0 ni—k edges, e ,• , 1 e2kl, e2k,2,• on the other hand, consists of edges, e 2 , 1 _1, 4 e4, = are the edges of G’ emerging from v, where ,2k. Let wjj be the other end of to be the plane (unsigned) graph consisting of , 2 e2, Define F (v) 0 star(v) if all the edges of G’ emerging from v have the same sign. If not, suppose ,• 1 , 2 e .signej,j = By defining , ,e 3 e3, 1 , 4 1,2,.•• ,2k) is the edge . Finally, F 1 wi, (G) and P(G) are 0 obtained from G’ by replacing each star(v) by Po(v) and I’(v) respectively. F (G) and 0 I’(G) are called the over-graph and under-graph of C, respectively. Example 4. Chapter 2. Kauffman bracket 27 AC (G) 0 F P(G) Figure 2.8: A link diagram D and related graphs G, P (G) and F(G) 0 Chapter 2. Kauffman bracket 28 Figure 2.9: An equivalent diagram In the link diagram D of Figure 2.8, each box represents some positive twists of crossing number two or more. If we shade the domains of the diagram, only three domains A, B and C are white domains. So its graph G has three vertices, labeled also by A, B and C respectively. Clearly, the graph has only negative edges except the top edge. Hence, from the definition, we get the graphs P (G) and P(G) as shown in Figure 0 2.8. A signed graph G is said to be semi-alternating if (1) 7 (G) 2 and (2) both F (G) and 0 P(G) are connected and non-separable. A link diagram D is said to be semi-alternating if the signed graph GD associated with D is semi-alternating. A link L is semi-alternating if L admits a semi-alternating diagram. K. Murasugi posed the following conjecture in [58] Conjecture. A semi-alternating diagram D is a minimal diagram of the link repre sented by D. In other words, the link diagram D realizes the minimal crossing number for the link. It is not difficult to check that the diagram in Example 4 is semi-alternating. However, Chapter 2. Kauffman bracket 29 the diagram is not minimal because it is obviously equivalent to the diagram in Figure 2.9. Hence the above conjecture is false. It is also in [58] that K. Murasugi raised the problem — whether a semi-alternating diagram is adequate. Let D be the diagram in Example 4. It is not difficult to see that D has minimal system {s}, where s takes value —1 only at the top vertex. By Corollary 3, D is not +subadequate. Heilce a semi-alternating diagram need not to be +subadequate, and we get a negative answer as well. Chapter 3 Jones knot invariants and Vassiliev invariants 3.1 Introduction This Chapter is divided into two parts. In the first part we investigate which knot in variant is Vassiliev invariant. In [13], J.S.Birman and X.S.Lin established a fundamental relationship between general Jones invariants and the Vassiliev invariants via a substi tution t &. We show here that this relation can be rearranged into a different form (Theorem 6), probably more natural, from which one can prove that every derivative of a general Jones invariant, evaluated at 1, is a Vassiliev invariant. While Bar-Natan observed that every coefficient of the Conway polynomial is a Vas siliev invariant, it is a little bit surprising that every coefficient of the Jones polynomial is not a Vassiliev invariant. This new result is proved in Theorem 7 below. On the other hand we also prove that the Jones polynomial evaluated at any nonzero complex number, except the cubic roots of unity, is not of finite type. In the second part of this chapter, we study the question: which numerical knot invariant can be approximated by Vassiliev invariants? We prove that the space of nu merical invariants which can be approximated by Vassiliev invariants is a dual space of some vector space. Finally, we give an interesting property of Vassiliev invariants from which we can retrieve, in a uniform manner, many known results about these invariants. For definitions and notations in this chapter, we refer to [7], [12] or [13]. 30 Chapter 3. Jones knot invariants and Vassiliev invariants 31 K_ K 0 K Figure 3.10: Related knot diagrams 3.2 Relationship between quantum group invariants and Vassiliev invariants Let S’ be the unit circle in the complex plane with a given orientation. A singular knot of order n is a piecewise linear immersion L: S 1 —+ 3 which has exactly n transverse 1R double points. Two singular knots L and L’ are equivalent if there exists an isotopy —+ [0, 1] such that h 0 R, t = id, h L 1 = L’ and the double points of hL are all transverse for every t E [0, 1]. We denote K, K and K_ the singular knots identical outside a small ball around a crossing and different inside a ball as shown in the Figure 3.10. Let IC, be the set of equivalence classes of singular knots with exactly i double points. In particular, IC 0 = Let’s abbreviate K, the set of all knot types. 3 by K> 1 and denote the Q-vector space generated by a set A 3K U by Q(A). Then the n-th finite-type space F is defined as the vector space generated by the set K> 0 subject to the following relations (1) K = K — K_ for K 1 K> Chapter 3. Jones knot invariants and Vassiliev invariants K (2) = 0 for K 32 . 1 K> Any element in the dual space Hom(F, Q) of F, but not in Hom(F_ , Q) is called 1 a Vassiliev invariant of order n or finite type invariant of order n. Recall that a quantum group invariant or a generalized Jones invariant is a knot or link invariant obtained from a trace function on a”R-matrix representation” of the family of braid groups {Bn Theorem 6 Let J = 1,2,3•• .}; refer to [12] for more details. 00 ct be a quantum group invariant and = 00 mO j(m)(fl (t_1)m m. be the Taylor series of J at 1, where J(m)(l) denotes the m-th derivative evaluated at 1. Then the constant term is 1 and the coefficient of order m form ‘‘‘) of (t — 1) is a Vassiliev invariant 1. Proof. First we note that a Vassiliev invariant of order m times a nonzero constant is also a Vassiliev invariant of order m and that the sum of a Vassiliev invariant of order m, and a Vassiliev invariant of order m’ is a Vassiliev invariant of order less than or equal to max{m,m’}. It is easy to see that the constant term is J(0)(1) = J(1) = 1, so we assume m By the definition of derivative, we have J(m)(1) cn(n—1)...(n—(m—1)). fl —00 Since n(n — 1)... (n — (m — 1)) = — j)m_l ( 1<i<m—1 + ( + 1. Chapter 3. Jones knot invariants and Vassiliev invariants 33 we have (1) m J = — n=—00 ( 1zm—1 + (_1)m_l(m 1 + cn — 1)! n=—oo ex in J = On the other hand, substituting t can. n=—c. c,-t and expanding cx in Taylor series, we have 00 00 m 00 00 = c,e” = J= fl=—oo n=—co m=O 00 m ( >Z m=O n=—oo By a theorem of Birman and Lin, see [13, Theorem 1], the coefficient of m 00 ,, -.-- 1 Xm 00 = is a Vassiliev invariant of order m for every m 1. Therefore, by the remark at the beginning of this proof, J(m)(1), as a linear combination of Vassiliev invariants of order m, is a Vassiliev invariant of order less than or equal to m. Since there is only one term in the summation having order m, is a Vassiliev invariant of order m. The proof is completed. D Corollary 8 The m-th derivative of a quantum group invariant evaluated at 1 is a Vas siliev invariant of order m. In particular, the m-th derivative of the Jones polynomial, evaluated at 1, is a Vassiliev invariant of order m. Proof. Clear from the proof of the above theorem. Corollary 9 For every quantum group invariant J, J’(l) = 0 and J”(l) = , where a 2 av is a constant and v 2 is the first nontrivial Vassiliev invariant. Proof. The first equation follows from the fact that there are no nontrivial Vassiliev invariants of order 1. The second one follows from that there is only one nontrivial Vassiliev invariant v 2 of order less than or equal to 2, up to constant multiplication. D Chapter 3. Jones knot in variants and Vassiliev invariants 2n + 34 1 Figure 3.11: Torus knot of type (2, 2n + 1) Remark 9 For the Jones polynomial J, since J”(l)(trefoil)= —6, we get J”(l) It is known that the second coefficient of the Conway polynomial = L”(1) = = . 2 —6v —J”(1), hence we reprove the fact that v 2 is the second coefficient of the Conway polynomial. 3.3 Some known invariants which are not Vassiliev invariants Bar-Natan observed that every coefficient of the Conway polynomial is a Vassiliev invari ant. In contrast, we have Theorem 7 Every coefficient of the Jones polynomial is not of finite type. Proof. First we prove the constant term c 0 of the Jones polynomial J = ct’ is not of finite type. Notice that if the minimal degree of the Jones polynomial J(K) for a knot K is greater than zero, then co(K) = 21 K 0. Denote the torus knot of type (2, 2n + 1) by , see Figure 3.11. According to V.Jones [31], we have ii’ rz ‘ tn — 2n+1) 11 — 1 — 2“ t 43 2n+2 4 f-i — & — 2n+3 I 4 —j— & Chapter 3. Jones knot invariants and Vassiliev invariants 2n + 1 35 2n Figure 3.12: A singular torus knot Hence 21 (K = 0 for 0 c ) 1 and co(Ki) = 1. ii Now extend c 0 to singular knots as usual, i.e., (K) = co(K+) 0 c — co(K_). Let K 2 be the singular knot as shown in Figure 3.12, By the above observation, we have (_1)P(2jco(K f ( l_P)+l) = (_1)2n m) = 2 2 co(K = Since n is arbitrary, we see that c 0 Now we turn to Cr 1 0. P pzO is not of finite type. for r> 0. It is well known that for the trefoil knot K we have 4 — 3 J(K)=t-i-t . t Let 2 K’#K + i be the connected sum of K +i and r copies of K. By a property of the 2 J(Kr#K = J(KyJ(K ) 1 + Jones polynomial, we have 2 ). It follows that the minimal 21 degree of J (K # K ) 1 + 2 r + 1 for n 1, hence we have n+i) = 0 for n 2 Cr(Kr#K As before we extend Cr 1. to singular knots. Since Cr(K’#Ki) = Cr(K’) = 1, we have Th) 2 Cr(KT#K (_1)P(2jCr(Kr#K n ( 2 _p)+l) = (_1)2n p=o P 0. Chapter 3. Jones knot invariants and Vassiliev invariants 2n - 1 36 2n Figure 3.13: A singular torus knot with 2n double points Hence Cr is not of finite type. Applying the above argument to the mirror images of knots we can prove that also not of finite type for r Cr is 0. This completes the proof. Remark 10 Theorem 7 implies that there are infinitely many integer-valued invariants which are not Vassiliev invariants. We can have a even stronger result, see Corollary 13 and Corollary L below. Remark 11 Theorem 7 is also true for the HOMFLY polynomials of one variable. Theorem 8 The genus, the minimal (maximal) degree of the Jones polynomial and the span of the Jones polynomial are not of finite type. Proof . Let K 2 be as in Figure 3.13, and K 21 be the torus knot of type (2,2n + 1). Denote the genus of knot K by g(K), then we know that g(K ) 21 i)=nforn0andg(K_i)=0. 2 J(K Now extend g to singular knots by g(Kx) = g(K) — g(K). = minimal degree of Chapter 3. Jones knot in variants and Vassiliev invariants 37 Then we have ) 2 g(K = l)+l) ( 2 (_l)P (2fl)g(K P pO 2n—1 = P 0 pz 2n—1 \ fl) 2n—1 fl)p = (2n-1) (_i) = (2n_1)(_1)P() _(2fl_1)(_1)2fl(:n) - = = (-1) -(2n - (2;) + - (_i)2n l)(_l)2n(j (_i) (j 2n + (_1)2n()2n (l)2n(j LO. ) = 0, the fifth equality follows from Propo 1 The second equality holds because g(K sition 2 for k = 1 and 0. Since n is arbitrary, we have proved the theorem for genus and the minimal degree of the Jones polynomial. i) = n + 3, hence a 2 For the span of the Jones polynomial, we know that span(K similar argument can be applied. This completes the proof. Remark 12 For K 1 where n + 2 D 1, the unknotting number is n, the minimal crossing number is 2n + 1 and the braid index is 2. Hence the above proof also shows that these invariants are not of finite type. These facts are already known. See J.S.Birman [12]. Remark 13 Theorem 8 is also obtained by J. Dean [15], Rolland Trapp [72] and others. Chapter 3. Jones knot invariants and Vassiliev invariants 38 In connection with Theorem 6, we may ask if Theorem 6 holds for Taylor series expansion in powers of (t — c) with c 1. The answer is no because of the following theorem. Theorem 9 The Jones polynomial evaluated at any nonzero complex number, except 1, and w , is not of finite type, where w is the primitive root of x 2 3 = 1. Proof. As before we extend the Jones polynomial to singular knots. Let K 2 denote the singular knot as in the proof of Theorem 7. Since 1 = _2 (1 — — 3+) t 23 t we have ) 2 J(K = (—1) (2n) P p=o J(K _ ( 2 )+i) 2n = p=o = — — 1 -t 2 — 1 P {(-1) (i)P( 2 — () 2 t jt2n_P+3 P p=o {(l — 2 t) — (1 2 t 2 ) 3 t — (1 3 t = — = 1 = 1 — t2 2 (1 — — — t) ( 2 1 — t)(1 n)t32n_P+3 p — (1 3 2+t t) (1 + t + t 2 t (1 ) 2 — t2( H 2 3) (1) ( j 2 + E(_1)P( — 3+ t (1 2 i — } 2 ) 3 t — 21 ) 2 t + t + ). It follows that if t is not a root of the above term, then J(K ) is nonzero. Hence, to 2 prove the theorem we only need to deal with these roots. To this end we consider some other singular knots. Let K ’ be the knot as shown 2 in Figure 3.14. Chapter 3. Jones knot invariants and Vassiliev invariants 39 2n+1 2n+2 Figure 3.14: A singular torus knot with 2n + 1 double points Similar to the above computation, we obtain ) J(K ’ 2 — = 1 (1 t) ’ 2 — (1 t ) 3 1+t+t t ( 2 ). ) 2 1+ _t ( (1 +t+t 2 ’) and (1 2 ) 2 It is easy to see that —1 is the only common root of (1 —t ). Since t t ) t+2 = —1 is a zero of order 1 for (1 — (—1) J(K ) ’), 2 t — ) 1 +t+2 t ( 2 0. Therefore, given any to which is not a cubic root of unity, for every positive integer n there exists a singular knot K with more than n double points such that J(K)(to) 0. Hence the Jones polynomial evaluated at to is not of finite type. The proof is completed. D Remark 14 We do not know whether every numerical invariant can be approximated by Vassiliev invariants. However Theorem 9 together with Theorem 6 imply that there are infinitely many non-finite-type invariants which can be approximated by finite-type invariants. Remark 15 The Jones polynomial evaluated at any cubic root of unity is known to be the constant 1. Hence we have completely determined whether the Jones polynomial, evaluated at a complex number, is of finite type. Chapter 3. Jones knot invariants and Vassiliev invariants 40 Remark 16 A similar proof shows that the one-variable HOMFLY polynomials evaluated at a complex number outside the unit circle are not of finite type. 3.4 Numerical invariants which are approximable by Vassiliev invariants Recall that a numerical knot invariant is a map v : IC —+ Q. Let us restate the following —f Q, does there exist a sequence important problem described in [13]. Problem 1. Given any numerical invariant v IC of Vassiliev invariants {v : K; —+ Q i = 1,2,. lim v(K) = . .} such that v(K) for every K € IC? This problem can be posed in the following way: is the Vassiliev invariant space dense in the 0-th cohomology of the knot type space? Another closely related problem is Problem 2. Do Vassiliev invariants distinguish knots? We are going to study the closure of the Vassiliev invariants in the 0-th cohomology of the space of knot types. Some equivalent conditions will then be given which lead to a better understanding on Vassiliev invariants. Recall that the n-th finite type space F is defined as the vector space generated by the set IC> 0 subject to the following relations (1) K=K+—K_ (2) K = 0 for for 1 KeIC> K E K;>,i Chapter 3. Jones knot invariants and Vassiliev invariants Note that we could allow n = 41 in the definition. More precisely, F is the Q-vector ) 0 0 subject only to relation (1). Since Q(K> space generated by the set K> = ) 0 Q(K , and relation (1) simply kills the second summand, we have Q(K> ) 1 Proposition 6 F = . Q(K ) 0 It is clear that there is a sequence of finite type spaces and maps 1 F F ... ,‘ 0 F where each map is a natural and epimorphism. If we regard the sequence as an inverse system, We have a projective limit denoted by limF Clearly, for every finite-type space F, we have a natural epimorphism F —+ F such that the diagram 1 F_ commutes. Passing to limit, we have a homomorphism lifl1F. Now the problem arises whether Tt is injective. More generally, what is the dimension of the kernel of 7r? What kind of elements are in this kernel? Analogously, one can ask Chapter 3. whether it Jones knot invariants and Vassiliev invariants 42 is surjective. If not, what is the dimension of its cokernel? we will answer some of these questions, which, as we can see, are very closely related to the Vassiliev invariants. Let us denote the Vassiliev space of order n by V = Hom(F, Q). Then it is clear that 0 C 1 V V •••C V C V+ C C •••C VC Hom(M,Q) C H 1 where V = U V, M is the image of F under it and H = Hom(F, Q) is the O-th cohomology group of the knot type space. Note that the inclusions are in the following sense: if v e V, , then v E Vi-, if and only if v factors through F, that is, if and only if 1 there exists a v’ V, such that the following diagram commutes 1 F+ F Hence V is the set of all Vassiliev invariants of order less than or equal to n. Let A C H. The closure of A in H, denoted by elements to f, f H such that there exists a sequence {f} A, is defined to be the set of all of A which pointwisely converges namely limf(K)=f(K) for every Ke)C. Then we have Theorem 10 V= Hom(M, Q), where M is the image of F under it. Chapter 3. 43 Jones knot invariants and Vassiliev invariants Proof. Let us first prove Vc Hom(M,Q). Consider F—M—-*•••-—F. and denote the composite of the above homomorphisms by d. Then we clearly have Ker(ir) C Ker(d) for every n. Suppose now f E V, then there exists a sequence 0, and hence f(Ker(7r)) = V,,, for every n. It follows that f,-, Without loss of generality, we may assume that = of V such that for every KEIC. limf(K)=f(K) f(Ker(d)) {f} = 0. This means M such that f(xo) 0. It follows 0. Therefore, f(Ker(ir)) that E Hom(M,Q). f Now we prove the other inclusion. Suppose f:M—Q 0 is a nontrivial fullctional. Then there exists an x that f is surjective, hence we have M , 2 ,x 1 Choose x 0 M. Since x , xj, = Ker(f) e Q(xo). , 2 ,x 0 ,x 1 to be a basis for Ker(f), then x ,x,••• is a basis for 0 is not zero in F. Since d 0, there exists an n such that the image of x , x, 0 ,• 1 is surjective and F is finite dimensional, we can choose {x 1 <m 2 image of this set is a basis for F and n < n, with r x,} such that the such that for any x not appearing in the set, d(x) is a linear combination of d(x)’s with f:F—+Q , j < i. Define Chapter 3. Jones knot invariants and Vassiliev invariants by taking 0 f(d(x ) ) = f(xo) and f(x) Note that dm(XO) f. in some 0 for every i. 0 in Fm for every m Vm in a similar way for every m converges to = 44 n, and therefore we can define fm E n. We claim that the sequence In fact, any finite set {x ,x 0 ,•• 1 ,x,}. But then it appears in , {f} pointwisely x} must, by construction, appear {XO,Xmi, ,Xmr} for every m n. Hence we have 1imf(x) This proves our claim, in other words, f(x). f V. D Corollary 10 The answer to Problem 1 is positive if and only if Ker(ir) is trivial. Corollary 11 The answer to Problem 2 is negative if and only if there are two different knots K and K’ such that K — K’ Ker(ir). Proof. If the Vassiliev invariants do not distinguish two knots K and K’, then for every Vassiliev invariant v, we have v(K) the definition of projective limit, K = — v(K’), and hence, K K’ e K’ in F for every n. By Ker(ir). Conversely, if K—K’ E Ker(r), we clearly have that v(K) = v(K’) for every Vassiliev invariant v. This completes the proof. 3.5 D Knot connected sum in finite type spaces Let K be a knot in R . Define 3 K#:C—K by taking K#(K’) = K’#K. It induces a homomorphism Q() Q() ,‘ Q(io) ,‘ F. Chapter 3. Jones knot invariants and Vassiliev invariants 45 Extend K# to singular knots as follows K#(K) = K#(K) — K#(K). It is not difficult to see that K induces a homomorphism, still denoted by K, F —+ F. Proposition 7 K is an isomorphism Proof. This is a corollary of Gussarov’s result, see [23] and [60]. Or one can also argue as in the proof of Proposition 10 below. It is easy to see that the diagram commutes, and that K# induces a homomorphism, still denoted by K#, lim F It is again an isomorphism. Notice that the diagram —+ lim F. Chapter 3. Jones knot invariants and Vassiliev invariants K# Fc,D 46 00 F 1 F commutes, and by passing to limit we get the following commutative diagram F K# M M It follows that K#(Ker(7r)) C Ker(’r). This property is useful in the proof of the following theorem. We say that K and K’ are distinguishable by Vassiliev invariants if there exists a Vassiliev invariant v such that v(K) v(K’). K and K’ are weakly distinguishable by Vassiliev invariants if there exist a Vassiliev invariant v and a knot L such that v(K#L) v(K’#L). Based on the same observation preceding Theorem 10, we may now prove a property on Vassiliev invariants similar to that of the quantum group invariants. Theorem 11 K and K’ are distinguishable by Vassiliev invariants if and only if they are weakly distinguishable by Vassiliev invariants. Chapter 3. Jones knot invariants and Vassiliev invariants 47 Proof. One direction is clear. Suppose there exists a Vassiliev invariant v and a knot L such that v(K#L) Theorem 10, K — v(K’#L). If v(K) = v(K’) for every Vassiliev invariant v, by K’ E Ker(ir), hence L#(K every knot L. By Theorem 10 again, v(K#L) — = K’) = K#L — K’#L € Ker(ir) for v(K’#L) for every knot L. This is a contradiction. The proof is completed. 0 K# is always injective because it is an isomorphism on the projective limit. The following theorem shows that this homomorphism reflects some important properties of Vassiliev invariants. Theorem 12 K—K’€ Ker(ir) if and only if K#=K :M—4M Proof. Suppose K# Conversely, let K — K, then K = e K’ = = L#K Therefore, K#(L) = K(L). Conjecture. K# = K : M K#() = K’ in F for every n, therefore, K—K’ E Ker(7r). Ker(ir), then K an arbitrary knot, then L#(K) K#(L) = = = K’ in F for every n. For fixed n, let L be L#(K’). Hence L#(K) = L#(K’) = L#K’ = K(L). 0 —+ M if and only if K#ir(1() Kir(K) if and only if K(M). The basis for proposing this conjecture is the following proposition. Proposition 8 If Problem 1 has a positive answer, then the above conjecture is also true. If problem 2 has a positive answer, then the first part of the conjecture is true. Proof. Clearly we have K# = K K&7r(AC) = Kir(K) K(M) = K(M). Chapter 3. Jones knot invariants and Vassiliev invariants Now suppose that there are Kand K’ with K# K#(M) knot L = e K(M). Since K#(K) K ( 48 hence K K’ ) such that I(PC), we may, without loss of generality, choose a K#(K) but not in K(K). Note that the images of Q(K#(K)) and Q(K4(K)) under ir are the same because K#(M) Q(K(K)) such that r(x) hence Ker(ir) = K(M). Hence there exists an element x E ir(L). Clearly, x L in F, therefore 7t is not one to one, {0}. The proposition now follows from Corollary 10. The second conclusion of this proposition is shown in the same way. 3.6 0 Some properties of Vassiliev invariants The following theorem is an interesting observation which allows us to retrieve, in a uniform manner, some known interesting results about Vassiliev invariants. Theorem 13 Let K be a knot which is equal to the unknot in F. Then for every knot L, K#L = L in F, where K denotes the connected sum of i copies of K. Proof. Since K is equal to the unknot in F, K# K(L) = = id on F. Therefore, 1 K # L L for every positive integer i. = 0 According to Gusarov and Yamamoto (see [60], for example), there exists a nontrivial knot K which is equal to the unknot in F. By Theorem 13, K#L’s have the same Vassiliev invariants. Since K#L’s are clearly different knots, we conclude that Corollary 12 (Lin and Stanford) There are infinitely many knots sharing the same Vas siliev invariants of order < n. Corollary 13 Let v be a Vassiliev invariant of order n. Then for every x E 1v ( 1 x) fl K = 0 or oc. Q, Chapter 3. Jones knot invariants and Vassiliev invariants Proof. If v’(x) fl K 49 0, choose a knot L such that v(L) = x. Let K be the knot which is equal to the unknot in F. By Theorem 13, v(K#L) = v(L) for every i. Hence the corollary. D Corollary 14 Numerical invariants such as the minimum crossing number, the unknot ting number, the braid index, the numerical invariants with finite support, etc are not Vassiliev invariants. Proof. The minimum crossing number of a knot is zero if and only if the knot is the unknot. Therefore, by the above corollary, the minimum crossing number is not a Vassiliev invariant. Similarly, we can prove the other cases. D Corollary 15 (Trapp) The complement of the Vassiliev invariants is dense in the 0-th cohomology of the knot type space. Proof. Clearly, every numerical invariant can be approximated by numerical invariants with finite support. By the above corollary, these invariants are in the complement of the Vassiliev invariants, hence the corollary. Theorem 14 If one of the numerical invariants listed in Corollary 14 can be approxi mated by Vassiliev invariants, then the Vassiliev invariants distinguish nontrivial knots from the unknot, in other words, there exists no nontrivial knot K such that K — U E Ker(ir) for the unknot U. Proof. If there is a nontrivial knot K such that K — U E Ker(Tr), then using a proof similar to the proof of Corollary 13, we can show that every numerical invariant v E V has the property described in Corollary 13. But invariants listed in Corollary 14 do not Chapter 3. Jones knot invariants and Vassiliev invariants 50 have this property, thus they are uot approximable by Vassiliev invariants, contradicting the hypothesis. Finally let us make a few comments on r. While we do not know whether KerQir) = 0, it is not difficult to see that if the kernel is nontrivial then it is infinite dimensional. The cokernel of it, on the other hand, has uncountable infinite dimension. So far we do not know what role is played by the cokernel of ruled out. it in knot theory, but its interest cannot be Chapter 4 Finite type invariants for lens spaces 4.1 Introduction Let M be a connected and oriented 3-manifold, S’ be the unit circle in the complex plane with a given orientation. A singular knot of order n is a piecewise linear map L : S’ —+ M which has exactly n transverse double points. Two singular knots L and L’ are equivalent if there exists an isotopy h : M = id, h L 1 —+ M, t E [0, 1] such that L’ and the double points of hL are all transverse for every t E [0, We denote by 1]. the set of equivalence classes of singular knots with exactly n double points. Clearly, £ = £° is the set of all knot types. Since L is oriented, for each double point we have two resolutions, denoted by L+ and see Figure 3.10. For more details, see [46]. Let C be the set of homotopy classes of loops in M, i.e. the set of conjugacy classes of iri(M). Then it is easy to see that every singular knot L belongs to a unique loop homotopy class; moreover, if L is equivalent to L’, then L and L’ belong to the same loop homotopy class. Let R be a ring with unit, and R(A) be the free module over R generated by a set A. Define F(M) where = is the submodule generated by the following relations (1) L = L — L_, L E £ 51 for some i Chapter 4. Finite type invariants for lens spaces 52 n+1 L(x (2) = (n+1) 0, L F(M) is called the n-th finite type space of M. The dual space of F(M), denoted by V,,(M), is called the Vassiliev space of M, and elements of V(M) are called finite type invariants or Vassiliev invariants. For M = , we already know much about V(IR 3 R ). Refer to [7], [12], [13] and [46] 3 for more details. But we know little about V(M) for a general 3-manifold M. X.S. Lin proved in [46] that V(M) = ) when 7ri(M) 3 V(IR = (M) 2 ir = 0. We are still looking forward to more developments along this direction. Since the solid torus plays a very important role in the study of 3-manifolds, it is our goal to understand more about V(M) for a solid torus. In this chapter, we first study the finite type spaces for a solid torus, then show that the Vassiliev spaces of lens spaces L(p, q) and L(p’, q’) are isomophic if p = p’. Consequently, we see that Vassiliev spaces do not distinguish manifolds. 4.2 An operation on finite type spaces Let c C, and £)(M) = {L E E c}, then define F,(M) where .— = R())/ 0 > is generated by the following relations (1) = L — L_, L E £ for some i n+1 (2) Proposition 9 F(M) LcE’Ex = eCEC 0, L E F(M). Proof. It is easy to see that if L follows. = c then L and L_ are also in c. Hence the proposition D Chapter 4. Finite type invariants for lens spaces Corollary 16 V(M) = 53 fJcV,(M), where V(M) is the dnal space of F(M). It follows that the study of F(M) is reduced to the study of each F,(M) which is still a complicated task. Let M = 3 and K be a knot in F. Define R K#C—÷L by taking K(L) = K#L. This induces a homomorphism R(C) - R(L) —÷ —÷ . F,(R ) 3 Extending K# to singular knots by K#(L) = K#(L+) — K#(L_), we can define a homomorphism, still denoted by K#, ) 3 K# : F(1 —+ . F(IR ) 3 By Proposition 7, K# is an isomorphism. , for example, 3 Let us now study the case when the manifold M can be embedded in 1R handle bodies, Whitehead manifolds, etc. Let 3 M—-R be an embedding, and c E C be a loop homotopy class. Given K c, choose a 3-ball inside of M, denoted by B, such that K fl B is a simple line segment. We try to construct a map K# : £(B) For each L —k £(B), let K#(L) be a “connected sum” of K and L. Here the “connected sum” is performed inside the ball B. Strictly speaking, we understand the “connected Chapter 4. Finite type invariants for lens spaces 54 sum” as follows: for each knot type, choose a representative knot L and a band connecting one point on L with a point on B fl K, then do the connected sum along the band. Since B — K U L is path connected, we always assume the band is contained in B. In this way we get a map from £(B) to £(M), hence a map to F(M), also denoted by K#. Now extend K# to F(B) via the formula K#(L) K(L) = K#(L_), — and obtain homomorphisms F,(M) F(B) -- , F(R ) 3 where i. is induced by inclusion map i. It follows from Proposition 7 that the composition of the above homomorphisms is an isomorphism. Theorem 15 F,(M) = ) 3 F(R ker(i). Proof. By the above remark, we have an exact sequence 0 —* Ker(i) —+ F(M) -- . F(R ) 3 ) 3 This sequence splits because we have a homomorphism K#(i*K#) 1 : F(IR such that iI(iK#)’ = Fc(M) id. Therefore we have F,c(M) = ) 3 F(R ker(i). Let us study the special case when M is the solid torus T. Since C this case, we have F(T) and —+ = CEZF(T). = iri(M) = Z in 55 Chapter 4. Finite type invariants for lens spaces Connected sum of two knots Two knots in a solid torus Figure 4.15: Connected sum of two knots in solid torus Corollary 17 For every c € Z, F,c(T) = ) 3 F(IR e ker(i). sum” for Similar to the “connected sum” described before, we have a “connected knots in T. We describe it by diagrams. See Figure 4.15. the “con Note that the orientations should coincide with each other. Note also that and hence nected sum” is dependent on the band on which the surgery is performed, might be noncommutative and nonassociative. d as Now that each pair of knots K, K’ can be assigned a “connected sum”, denote T simply by usual by K#K’, we can extend such “connected sum” to singular knots in LX#L’ L+#L’ — L#L’. on one can Note that L #L’ may be represented by a singular knot. With this operati easily prove the following Theorem 16 For n 1, F(T) CEzF,(T) is a graded algebra (possibly nonassocia tive) with the unknot serving as unit element. ld M. Remark 17 A similar “connected sum” can be introduced for any manifo 56 Chapter 4. Finite type invariants for lens spaces Choose a “connected sum” as above, and let K be a knot. Define K#:L—4C by taking K#(L) = K#L. This induces a homomorphism R(C) R() , , F(T). Extend K# to singular knots as follows: K#(L) = K#(L+) — K#(L_). It is not difficult to see that I induces a homomorphism, still denoted by K#, K# : F(T) —f F(T). Proposition 10 K# is an isomorphism. Proof. First we assume that the knot K has winding number zero. Then K# takes F(T) into F(T), and we only need to show that K# restricted to F is an isomorphism for every c E Z. For n = 0, F(T) is a space of dimension one generated by a knot, say L, which is in c. K# carries L to K#L which is equal to L in F°(T) because K can be changed into the unknot via crossing changes. Hence the proposition is true in this case. Assume the proposition true for values diagram < n and consider the following commutative Chapter 4. Finite type invariants for lens spaces 57 0 F,(T) (T) 1 F I K . K F(T) (T) 1 F The bottom K# is an isomorphism by induction assumption. The top K# is an identity map because K#L can be changed into L via crossing changes. Therefore, by the five lemma, the middle K 3 is also an isomorphism. Now we assume that K has winding number r. Choose a knot K’ such that K’ has winding number —r. For example, we may choose K’ to be the knot obtained from K by reversing the orientation of K. Then K#K’ has winding number zero, hence (K#K’)# is an isomorphism by the previous argument. Moreover, this is indeed true for any chosen “connected sum”. Hence the composite I K is also an isomorphism. Likewise for K Ii. Therefore K# is an isomorphism. The proof is completed. 4.3 0 Similarity classes Let us recall the definition of similarity classes. Following [46], let L : S 1 —+ M be a singular knot. The pre-image of a double point consists of two points in S. Identi fying these two points into one for each double point, we get a 1-dimensional compact Chapter 4. Finite type invariants for lens spaces polyhedron, denoted by FL. 58 Let L and L’ be singular knots of order n in M. We say that L is similar to L’ if (1) there is a homeomorphism FL —+ FL’ which lifts to an orientation preserving homeomorphism 31 S’, and (2) there is a homotopy S’ —+ M, t E [0, 1], with 4o parametric values 0 e • £‘), 1 < < t i = = <tj,, < L, 4 = L’, and such that there are finitely many 1 so that 1,... ,k; • for different t’s in the same connected componemt of [0, 1] are equivalent singular knots in — {t , 1 . tk}, the ‘s and • when t passes through t, q changes from one resolution of to the other. The equivalence classes of singular knots so defined are called similarity classes. Choose a representative from each similarity class to form a set Proposition 11 F,(M) is generated by Proof. For n = where Q = . {L E 1ZL E c}. 0, all the knots in c are similar. We can choose a knot as the representa tive so that F(M) is generated by the chosen knot, hence Proposition 11 is true in this case. To prove that it is also true for n, under the induction assumption that it is true for < n — 1, we consider the exact sequence R(Q) Let x E F(M) and x’ = -- F,(M) --* (M) 1 F —f 0 ?r(x). By the induction assumption x’ can be written in the form = L 3 >a with oniy finitely many as’s nonzero. 3 E U<_ L 1 Chapter 4. Finite type invariants for lens spaces Regarding ZaL as an element in F(M), we have x 59 — ZaL = g(y) for some y E R(Qj. Since y is a linear combination of elements of f, so is g(y). Hence x is a linear combination of elements of U Q. The proof is completed. 0 D It is clear that similarity of L and L’ in T implies their similarity in R . Let’s fix a 3 similarity class L in R and ask how many similarity classes in T correspond to L. If there were only finitely many similarity classes for each L, then F,(T) would be finitely generated. Unfortunately, this isn’t quite the case. This is because one can always let one arc go clockwise around T p times and let another arc go counterclockwise around T q times so that p similarity classes. — q = c, while different values of p and q may correspond to different In other words one can easily construct infinitely many similarity classes for every c. On the other hand, in a slightly different way, we will define a finitely generated module which is important to the study of the finite type space of lens spaces. Given a knot L, assume that all the double points are in the upper part of the solid torus, or equivalently in a given 3-ball in T so that outside of the ball, the knot consists of parallel strings. Such a singular knot embedding is called a special embedding. It is clear that every singular knot in T has a special embedding. Now delete all double points of a special embedding for the singular knot to get several arcs, labelled as l(i = 1, . . k). For each i, we choose an oriented arc lj in the upper part of the solid torus so that the beginning point 1 is the end point of and the end point of i is the beginning point of I and so that ij has no other intersection with L. Then the composite I * i, is a loop in the solid torus which represents an element in the homology group H (T) 1 = Z. This element, an integer, is called the “winding number” of I with respect to the special embedding. , 1 For every sequence a , ak of integers, one can construct a knot L’ such that the corresponding winding numbers are a , 1 4.16 shows. , as Figure 3 ak and L is similar to L’ in R Chapter 4. Finite type invariants for lens spaces 60 3 times -1 times Figure 4.16: Arc winding numbers It follows from the following proposition that any singular knot which is similar to L in R 3 can be constructed in this way. As noted before there are infinitely many such knots that are not similar to each other in T. Proposition 12 Let L be a singular knot contained in a 3-ball in T. Then any singular 3 to L is similar in T to a singular knot constructed knot K in T which is similar in 1R from L via letting l(1 i k) go around T a 1 times , where a 2 is the winding number of an arc of K which corresponds to i for some special embedding of K. Chapter 4. Finite type invariants for lens spaces 61 Proof. We may assume that all the double points of L and K are in the upper part of the solid torus. Since L and K are similar in R , we can move the double points 3 of L via an isotopy of T to match the double points of K according to the one to one correspondence determined by the similarity property. Hence we may assume that L is already in this position. Now we let each arc l go around the solid torus a times , where a is the winding number of the corresponding arc of K. We denote the resulting singular knot by L’. We need to prove that K is similar to L’ in T. The corresponding arcs of K and L’ constitute a loop which is homotopically trivial because their winding numbers are the same. Therefore the arcs of K are homotopic to the corresponding arcs of L’ relative to the end points. By Denn’s lemma, every corresponding arc pair bounds a disk. Perturbing this disk if neccesary, we can ensure that it avoids the double points and intersects with knots K and L’ transversely. With such disks, one can easily construct the required homotopy. The proof is completed. For a given integer p D 0, one can introduce a relation between singular knots in T. Namely, L and L’ are said to be p-related if one of them can be obtained from the other by taking connected sum with some knot of winding number +p. The relation generated by p-relation and similarity relation is called p-similarity. Modulo the p-relation in the finite type space F(T), we have Proposition 13 F(T)/ = Z{F(T)/ ‘-}, where is the p-relation. Moreover, this module is finitely generated. Proof. Since F(T) = zF(T), it is clear that if two singular knots are p-similar, the difference of their grades in F(T) must be a multiple of p and that any element of F(T) is p-similar to some element of F,0(T) (simply let one string go ±p times around the solid torus T). Also notice that there are some identifications within the same summands of F(T). Therefore the first part of the proposition is true. Chapter 4. Finite type invariants for lens spaces 62 For the second part, we need to note that F(T) is generated by the set of repre sentatives from similarity classes of order less than or equal to n , hence F(T)/ ‘-‘.‘ is generated by the set of representatives from p-similarity classes in T of order less than or equal to n. Since the set of similarity classes in R 3 of order less than or equal to n is a finite set, it follows from Proposition 5 that the set of p- similarity classes is a finite set too ( of cardinality not excceding the number of similarity classes in R 3 times Hence the second part of the proposition. 4.4 n 2 p ) 0 Finite type spaces for lens spaces Now we turn to study lens spaces and show that their finite type spaces can be described by the finite type space of the solid torus. Let L(p, q) be a lens space. Then it can be obtained by attaching a 2-handle to the solid torus T along a torus knot of type (p, q) followed by attaching a 3-handle. From this point of view, it is not difficult to see that every singular knot is isotopic in L(p, q) to a singular knot contained in the solid torus. Therefore we can define F(L(p, q)) using only those singular knots contained in T. Notice that adding a 2-handle is to add some relation to the module F(T) = eF(T) and adding a 3-handle does not affect the module. Hence, to determine the finite type space of L(p, q), we only need to figure out the relation corresponding to the 2-handle. The relation can be described as follows. Let L and L’ be two singular knots such that L’ can be obtained from L by sliding a string of L over the attached disk. We say that L’ is a sliding of L in this case. The relation is simply the one generated by slidings. Lemma 8 Sliding a string over the attached disk is equivalent to taking connected sum with an oriented torus knot of type (p, q), where the orientations may be different for Chapter 4. Finite type invariants for lens spaces 63 different slidings. Sliding a double point over the attached disk is equivalent to taking twice connected sum with an oriented torus knot of type (p,q). Proof. Since the boundary of the attached disk is the torus knot of type (p, q), sliding a string over the attached disk is equivalent to taking connected sum with an oriented torus knot of type (p, q). Sliding a double point over the attached disk can be realized by first “parallelly” moving two strings near the double point over the attached disk, then moving the double point along the band bounded by the two strings to where the double point is located after the double point sliding. Since the second moving is clearly an isotopy of the solid torus T, it is easy to see that sliding a double pint over the attached disk is equivalent to taking twice connected sum with an oriented torus knot of type (p,q). D Theorem 17 Two singular knots in T are similar in L(p, q) if and only if they are p-similar. Proof. Let L and L’ be two singular knots in T which are similar in L(p, q). Then L’ can be obtained from L via crossing changes, slidings and isotopies which are performed outside of the 3-handle. Since the torus knot of type (p, q) has winding number +p, by Lemma 8, slidings do not change p-similarity. Crossing changes and isotopies clearly do not change p—similarity. Hence, L and L’ are p-similar. This proves one implication. For the reversed implication, assume that L and L’ are p-similar. Notice that any knot of winding number +p is similar to the (p, q)-type torus knot in T. Hence a singular knot connected sum with a knot of winding number +p is similar to the same knot connected sum with the (p, q)-type torus knot. Hence L is related to L’ by crossing changes, isotopies and slidings. In other words, L and L’ are similar in L(p, q). The proof is completed. D Now we are going to prove the main theorem of this chapter. Chapter 4. Finite type invariants for lens spaces Theorem 18 F(L(p,q)) 64 where is the p-relation. Moreover, the finite type spaces of lens spaces are finitely generated. Proof. We know F(T) EEZF(T). Since “sliding” is equivalent to “connected sum”, for any singular knot K e n E Z, there exists a singular knot K’, obtained from K by taking iterated connected sum with a (p, q)-type torus knot, such that K’ E i with 0 < i <p — F(L(p,q)) where the relation q F,(T)/ Jq is generated by slidings with respect to the torus knot of type (p, q). Since the relation i: 1. Hence we have F,!(T)/ is contained in the relation q ‘-‘-‘, we have a quotient map for every i. Now we prove the theorem by induction on n. Since every F j(T) is one dimensional, 1 the theorem is clear for n 0. Assume it is true for n diagram: F(T)/q I I I — 1, and consider the following Chapter 4. Finite type invariants for lens spaces Here i 65 denotes the quotient map, and R(L4) is a submodule of F,(T) generated by similarity classes of order n in i. By Theorem 17, the relation p-relation. 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On Jones knot invariants and Vassiliev invariants Zhu, Jun 1995
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Title | On Jones knot invariants and Vassiliev invariants |
Creator |
Zhu, Jun |
Date Issued | 1995 |
Description | The main objective of this thesis is to study invariants of knots and links. First, a minimal system is associated to each link diagram D. This minimal system is an “an tichain” in the partially ordered set of all possible functions from the crossings of D to {—1, 1}. Such a system is then shown to be an effective tool for determining the highest degree as well as the span of the Kauffman bracket (D). When applied to a diagram D with n crossings which is “dealternator connected” and “rn-alternating”, the upper bound span (D) ≤ 4(n—m) is obtained. This is a best possible result providing a negative answer to the conjecture posed by C.Adams et al. Next, some examples are presented to show that a semi-alternating diagram need not be minimal. This disproves a conjecture posed by K.Murasugi. For knots and links, if J(t) is a so-called generalized Jones invariant, then J(n)(1), the n-th derivative of J(t) evaluated at t = 1, is a Vassiliev invariant. On the other hand, while the coefficients of the classical Conway polynomial are Vassiliev invariants, the coefficients of the Jones polynomial are now shown not to be Vassiliev invariants. An interesting property of Vassiliev invariants is then given to prove many known results in a uniform manner. It is known that all the Vassiliev invariants with order less than or equal to n form a vector space called the n-th Vassiliev space. It turns out that two lens spaces have the same Vassiliev spaces if and only if their fundamental groups are isomorphic. Consequently, Vassiliev spaces do not distinguish manifolds. |
Extent | 1452780 bytes |
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Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-06-05 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080010 |
URI | http://hdl.handle.net/2429/8830 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1995-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
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