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On Jones knot invariants and Vassiliev invariants

University of British Columbia

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.

Thesis/Dissertation

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2009-06-05

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http://hdl.handle.net/2429/8830

Mathematics

University of British Columbia

UBCV

DSpace