A remark on a finiteness of purely cosmetic surgeries

Abstract

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 $g(K)\ge \frac{3}{2}b(K)$, 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.

5 pages, no Figure

References in corpus (3)

• The Jones polynomial and graphs on surfaces
• 3-braid knots do not admit purely cosmetic surgeries
• A quantitative Birman-Menasco finiteness theorem and its application to crossing number