A splicing formula for the LMO invariant

TL;DR

This paper establishes a splicing formula for the LMO invariant of rational homology 3-spheres, expressing it in terms of invariants of simpler components, and recovers known results in special cases.

Contribution

It introduces a new splicing formula for the LMO invariant, extending previous work and providing a satellite formula for the Kontsevich-LMO invariant.

Findings

Recover Fujita's formula for the Casson-Walker invariant in low degrees

Second term of the Ohtsuki series is not additive under splicing

Splicing formula applies to links and knots, enabling satellite invariants

Abstract

We prove a "splicing formula" for the LMO invariant, which is the universal finite-type invariant of rational homology $3$-spheres. Specifically, if a rational homology $3$-sphere $M$ is obtained by gluing the exteriors of two framed knots $K_1 \subset M_1$ and $K_2\subset M_2$ in rational homology $3$-spheres, our formula expresses the LMO invariant of $M$ in terms of the Kontsevich-LMO invariants of $(M_1,K_1)$ and $(M_2,K_2)$. The proof uses the techniques that Bar-Natan and Lawrence developed to obtain a rational surgery formula for the LMO invariant. In low degrees, we recover Fujita's formula for the Casson-Walker invariant and we observe that the second term of the Ohtsuki series is not additive under "standard" splicing. The splicing formula also works when each $M_i$ comes with a link $L_i$ in addition to the knot $K_i$, hence we get a "satellite formula" for the Kontsevich-LMO invariant.