A simple formula for the Casson-Walker invariant

Details A simple formula for the Casson-Walker invariant A simple formula for the Casson-Walker invariant

2008-11-04

arxiv.org/abs/0811.0606v2

J. Knot Theory and Ramif. 18 (2009) 841--864

Mathematics

Geometric Topology

22 pages, 17 figures

Scientific paper

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.

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