Cosmetic two‐strand twists on fibered knots

Abstract

Let K be a knot in a rational homology sphere M. This paper investigates the question of when modifying K by adding m > 0 half-twists to two oppositely oriented strands, while keeping the rest of K fixed, produces a knot isotopic to K. Such a two-strand twist of order m, as we define it, is a generalized crossing change when m is even and a non-coherent band surgery when m = ±1. A cosmetic two-strand twist on K is a non-nugatory one that produces an isotopic knot. We prove that fibered knots in M admit no cosmetic generalized crossing changes. Further, we show that if K is fibered, then a two-strand twist of odd order m that is determined by a separating arc in a fiber surface for K can only be cosmetic if m = ±1. After proving these theorems, we further investigate cosmetic two-strand twists of odd orders. Through two examples, we find that the second theorem above becomes false if 'separating' is removed, and that a key technical proposition fails when the order equals 1. A closer look at an order-one example, an instance of cosmetic band surgery on the unknot, reveals it to be nearly trivial in a sense that we name weakly nugatory. We correct the technical proposition to obtain a means of using double branched covers to show that certain band surgeries are weakly nugatory. As an application, we prove that every cosmetic band surgery on the unknot is of this type. This paper relies extensively on color figures. Some references to color may not be meaningful in the printed version, and we refer the reader to the online version which includes the color figures.

Cosmetic operations and Khovanov multicurves

“…In fact, Theorem 1.2 immediately implies the Cosmetic Crossing Conjecture for the unknot, originally due to Scharlemann-Thompson [ST89]. This conjecture is still open in general, but has been established for knots which are two-bridge [Tor99] or fibered [Kal12,Rog20], as well as for large classes of knots which are genus one [BFKP12,Ito21] or alternating [LM17]. In the second half of this paper, we illustrate the utility of this theory by giving elementary proofs of two other known results about the Cosmetic Crossing Conjecture.…”