A splicing formula for the LMO invariant. (20th December 2021)

• Record Type: Journal Article
• Authors: Massuyeau, Gwénaël; Moussard, Delphine
• Is Part Of: Canadian journal of mathematics. Volume 73:Number 6(2021)
• Page Start: 1743
• Page End: 1770
• Publication Date: 2021-12-20
• Subjects: 57M27; LMO invariant -- splicing -- homology sphere -- knot -- Kontsevich–LMO invariant -- Casson–Walker invariant -- surgery -- Jacobi diagram; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/S0008414X20000668
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: We prove a "splicing formula" for the LMO invariant, which is the universal finite-type invariant of rational homology three-spheres. Specifically, if a rational homology three-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 three-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.