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Register a new userA cabling formula for $ u^+$ invariant
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Authors
Zhongtao Wu
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We prove a cabling formula for the concordance invariant $ u^+$, defined by the author and Hom. This gives rise to a simple and effective 4-ball genus bound for many cable knots.
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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 $K_2subset M_2$ 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 Fujitas 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.
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