The Goeritz matrix and signature of a two bridge knot

TL;DR

This paper derives a formula for the signature of two bridge knots using the Goeritz matrix and continued fraction expansions, providing an algorithm to compute cases where the correction term vanishes.

Contribution

It introduces a method to compute the knot signature directly from continued fraction coefficients and identifies conditions for the correction term to be zero.

Findings

Signature formula for two bridge knots using Goeritz matrices

Identification of 'even continued fraction expansions' where correction term vanishes

Algorithm for finding suitable continued fraction expansions

Abstract

According to a formula by Gordon and Litherland, the signature of a knot K can be computed as the signature of a Goeritz matrix of K minus a suitable correction term, read off from the diagram of K. In this article, we consider the family of two bridge knots K(p/q) and compute the signature of their Goeritz matrices in terms of the coefficients of the continued fraction expansion of p/q. In many cases we also compute the value of the correction term. We show that for every two bridge knot K(p/q), there are "even continued fraction expansions" of p/q, for which the correction term vanishes, thereby fully computing the signature of K(p/q). We provide an algorithm for finding even continued fraction expansions. This article is the result of an REU study conducted by the first author under the direction of the second.