On Polynomial Invariants for Knots and Links

Author: Ho, Chi Fai

Year: 1986

Degree: Dissertation (Ph.D.)

Advisor: Fuller, F. Brock

Committee Members: Fuller, F. Brock; Anderson, Donald L.; Wilson, Richard M.; Wales, David B.

Option: Mathematics

DOI: 10.7907/2tfc-f172

Abstract

This thesis presents an investigation of many known polynomial invariants of knots and links. Following Alexander's original idea, we define another multi-indeterminant polynomial for links and show that it satisfies some of Torres' conditions. We conjecture that they are equivalent.

Conway polynomials have been known since the sixties. In this paper, we show that the polynomials of various orientations of a link are related, at least in the first and second coefficients. The relationship can be expressed as a function of the Conway polynomials of all sublinks.

A new invariant polynomial of knots and links has been discovered which is independent of the orientation. This polynomial is also invariant of link inverses. Moreover, it is different from the Conway polynomial and the newly discovered HOMFLY polynomial. It distinguishes the trivial 3-unlink and the Borremean ring of 3 components. Various properties of the polynomial are studied.

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