3-braid knots do not admit purely cosmetic surgeries

Cosmetic operations and Khovanov multicurves

“…In an entirely different direction, hyperbolic geometry techniques have been used to bound the lengths of cosmetic filling slopes [FPS21]. Using the strategies above and others, the conjecture has been established for knots of genus one, cables, connected sums and three-braids [Wan06,Tao19a,Tao19b,Var21].…”

Purely cosmetic surgeries and pretzel knots

“…The conjecture has been verified for 2-bridge knots [4], for connected sums [12], for 3-braid knots [14], for knots of Seifert genus one [16] and for prime knots with at most 16 crossings [3]. By the classification of Seifert fibered spaces, the conjecture also holds for torus knots.…”

A remark on a finiteness of purely cosmetic surgeries

“…Proof of Theorem 1. In the following we assume that b(K) ≥ 4 since Varvarezos proved the cosmetic surgery conjecture for the case b(K) = 3 [8]. Also, we assume that g(K) = 2.…”