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Register a new userA remark on a finiteness of purely cosmetic surgeries
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Authors
Tetsuya Ito
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By estimating the Turaev genus or the dealternation number, which leads to an estimate of knot floer thickness, in terms of the genus and the braid index, we show that a knot $K$ in $S^{3}$ does not admit purely cosmetic surgery whenever $g(K)geq frac{3}{2}b(K)$, where $g(K)$ and $b(K)$ denotes the genus and the braid index, respectively. In particular, this establishes a finiteness of purely cosmetic surgeries; for fixed $b$, all but finitely many knots with braid index $b$ satisfies the cosmetic surgery conjecture.
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We show that two Dehn surgeries on a knot $K$ never yield manifolds that are homeomorphic as oriented manifolds if $V_K(1) eq 0$ or $V_K(1) eq 0$. As an application, we verify the cosmetic surgery conjecture for all knots with no more than $11$ crossings except for three $10$-crossing knots and five $11$-crossing knots. We also compute the finite type invariant of order $3$ for two-bridge knots and Whitehead doubles, from which we prove several nonexistence results of purely cosmetic surgery.
Beliakova-Putyra-Wehrli studied various kinds of traces, in relation to annular Khovanov homology. In particular, to a graded algebra and a graded bimodule over it, they associate a quantum Hochschild homology of the algebra with coefficients in the bimodule, and use this to obtain a deformation of the annular Khovanov homology of a link. A spectral refinement of the resulting invariant was recently given by Akhmechet-Krushkal-Willis. In this short note we observe that quantum Hochschild homology is a composition of two familiar operations, and give a short proof that it gives an invariant of annular links, in some generality. Much of this is implicit in Beliakova-Putyra-Wehrlis work.
We show that all nontrivial members of the Kinoshita-Terasaka and Conway knot families satisfy the purely cosmetic surgery conjecture.
Let $Sigma$ be a surface of negative Euler characteristic and $S$ a generating set for $pi_1(Sigma,p)$ consisting of simple loops that are pairwise disjoint (except at $p$). We show that the word length with respect to $S$ of an element of $pi_1(Sigma,p)$ is given by its intersection number with a well-chosen collection of curves and arcs on $Sigma$. The same holds for the word length of (a free homotopy class of) an immersed curve on $Sigma$. As a consequence, we obtain the asymptotic growth of the number of immersed curves of bounded word length, as the length grows, in each mapping class group orbit.
Using the mapping cone of a rational surgery, we give several obstructions for Seifert fibered surgeries, including obstructions on the Alexander polynomial, the knot Floer homology, the surgery coefficient and the Seifert and four-ball genus of the knot.
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