A simple formula for the Casson-Walker invariant

TL;DR

This paper introduces simple Gauss diagram formulas for the Casson-Walker invariant of rational homology spheres, clarifying its dependence on link framing and providing skein relations and computational results.

Contribution

It provides the first straightforward Gauss diagram formulas for the Casson-Walker invariant, enhancing understanding of its dependence on link framing and enabling new computational techniques.

Findings

Gauss diagram formulas for the Casson-Walker invariant

Skein relations under crossing changes

Asymptotic behavior as framings tend to infinity

Abstract

Gauss diagram formulas are extensively used to study Vassiliev link invariants. Now we apply this approach to invariants of 3-manifolds, considering manifolds given by surgery on framed links in the 3-sphere. We study the lowest degree case - the celebrated Casson-Walker invariant of rational homology spheres. This paper is dedicated to a detailed treatment of 2-component links; a general case will be considered in a forthcoming paper. We present simple Gauss diagram formulas for the Casson-Walker invariant. This enables us to understand/separate its dependence on the unframed link and on the framings. We also obtain skein relations for the Casson-Walker invariant under crossing changes, and study its asymptotic behavior when framings tend to infinity. Finally, we present results of extensive computer calculations.