A remark on a finiteness of purely cosmetic surgeries

TL;DR

This paper proves that knots with genus at least 1.5 times their braid index do not admit purely cosmetic surgeries, establishing a finiteness result for such surgeries across knots with fixed braid index.

Contribution

It introduces a new inequality involving knot genus and braid index to determine the absence of purely cosmetic surgeries, advancing understanding of the cosmetic surgery conjecture.

Findings

Knots with genus ≥ 1.5 × braid index do not admit purely cosmetic surgeries.

Finiteness of purely cosmetic surgeries is established for knots with fixed braid index.

Most knots with a given braid index satisfy the cosmetic surgery conjecture.

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.