A splicing formula for the LMO invariant
| Autor: | Massuyeau, Gwenael, Moussard, Delphine |
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| Rok vydání: | 2020 |
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| Zdroj: | Canad. J. Math. 73:6 (2021) 1743-1770 |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 25 pages |
| Databáze: | arXiv |
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