A simple formula for the Casson-Walker invariant
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.
Cite
@article{arxiv.0811.0606,
title = {A simple formula for the Casson-Walker invariant},
author = {Sergei Matveev and Michael Polyak},
journal= {arXiv preprint arXiv:0811.0606},
year = {2011}
}
Comments
22 pages, 17 figures