Construction of an invariant for integral homology 3-spheres via completed Kauffman bracket skein algebras

TL;DR

This paper introduces a new invariant for integral homology 3-spheres derived from completed Kauffman bracket skein algebras, linking it to known invariants like Casson and Ohtsuki series.

Contribution

It constructs a novel invariant using completed skein algebras and Heegaard splittings, connecting it to finite type invariants and classical invariants.

Findings

The invariant recovers the Casson invariant as a coefficient.

For the Poincaré homology sphere, it matches the Ohtsuki series.

The invariant is a formal power series with topological significance.

Abstract

We construct an invariant $z (M) =1+a_1(A^4-1)+ a_2(A^4-1)^2+a_3(A^4-1)^3 + \cdots \in \mathbb{Q} [[A^4-1]]= \mathbb{Q} [[A+1]]$ for an integral homology $3$-sphere $M$ using a completed skein algebra and a Heegaard splitting. The invariant $z(M)\mathrm{mod} ((A+1)^{n+1}) $ is a finite type invariant of order $n$. In particular, $-a_1/6$ equals the Casson invariant. If $M$ is the Poincar\'{e} homology 3-sphere, $(z(M))_{|A^4 =q} \mod (q+1)^{14} $ is the Ohtsuki series for $M$.