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

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 Poincaré homology 3-sphere, $(z(M){)}_{|A^4=q}\mathrm{mod}(q+1{)}^{14}$ is the Ohtsuki series for $M$.

13 pages, 9 figures

References in corpus (2)

• The quotient of a Kauffman bracket skein algebra by the square of an augmentation ideal
• The Torelli group and the Kauffman bracket skein module

Cited by in corpus (2)

• The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera
• A formula for the action of Dehn twists on HOMFLY-PT skein modules and its applications