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