Rocky Mountain Journal of Mathematics

The Goeritz matrix and signature of a two bridge knot

Michael Gallaspy and Stanislav Jabuka

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Abstract

According to a formula by Gordon and Litherland \cite{GordonLitherland}, the signature $\sigma (K)$ of a knot $K$ can be computed as $\sigma (K) = \sigma (G) - \mu$, where $G$ is the Goeritz matrix of a projection $D$ of $K$ and $\mu$ is a ``correction term,'' read off from the projection $D$. In this article, we consider the family of two bridge knots $K_{p/q}$ and compute the signature of the Goeritz matrices of their ``standard projections,'' $D_{p/q}$, by explicitly diagonalizing the Goertiz matrix over the rationals. We give applications to signature computations and concordance questions.

Article information

Source
Rocky Mountain J. Math., Volume 45, Number 4 (2015), 1119-1145.

Dates
First available in Project Euclid: 2 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1446472426

Digital Object Identifier
doi:10.1216/RMJ-2015-45-4-1119

Mathematical Reviews number (MathSciNet)
MR3418186

Zentralblatt MATH identifier
1333.57011

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

Citation

Gallaspy, Michael; Jabuka, Stanislav. The Goeritz matrix and signature of a two bridge knot. Rocky Mountain J. Math. 45 (2015), no. 4, 1119--1145. doi:10.1216/RMJ-2015-45-4-1119. https://projecteuclid.org/euclid.rmjm/1446472426


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