The Goeritz matrix and signature of a two bridge knot

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.