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) ≥ 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.

On constraints for knots to admit chirally cosmetic surgeries and their calculations

“…Combining other constraints of cosmetic surgeries [IW,It1,De], the cosmetic surgery conjecture has been confirmed for many cases, such as, knots with at most 17 crossings [De], composite knots [Ta2], cable knots [Ta1], 2-bridge knots [IJMS], pretzel knots [SZ]. In fact, the number of purely cosmetic surgeries, even if exists, is finite in the following sense; for a given b > 0, there are only finitely many knots that admit a purely cosmetic surgery whose braid index is less than or equal to b [It3].…”