UBC Theses and Dissertations
Alexander invariants of links Bailey, James Leonard
In the three main sections of this thesis (chapters II, III, and IV; chapter I consists of definitions) we explore three methods of studying Alexander polynomials of links which are alternatives to Fox' free differential calculus. In chapter II we work directly with a presentation of the link group and show how to obtain a presentation for the Alexander invariant. From this we deduce that the order ideal of the Alexander invariant is principal for links of two or three components (the case of one component is well known) but nonprincipal in general for links of four or more components. In any event we show that only one determinant is needed to obtain the Alexander polynomial. In chapter III we use surgery techniques to characterize Alexander invariants of links of two components in terms of their presentation matrices. We then use this to show that the Torres conditions characterize link polynomials when the linking number of the two components is zero or both components are unknotted and the linking number is two. Chapter IV uses Seifert surfaces to prove a generalization of a theorem of Kidwell which relates the individual degrees of the Alexander polynomial to the linking complexity, to present an algorithm for calculating the Alexander polynomial of a two-bridge link from a two-bridge diagram and to prove a conjecture of Kidwell in the special case of two-bridge links. These results are then used to generate link polynomials from allowable pairs (a concept introduced in chapter III) and these results in turn are used to produce methods of generating allowable pairs.
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