2-strand twisting and knots with identical quantum knot homologies

Abstract

Given a knot, we ask how its Khovanov and Khovanov-Rozansky homologies change under the operation of introducing twists in a pair of strands. We obtain long exact sequences in homology and further algebraic structure which is then used to derive topological and computational results. Two of our applications include giving a new way to generate arbitrary numbers of knots with isomorphic homologies and finding an infinite number of mutant knot pairs with isomorphic reduced homologies.

v2: Main results unchanged. More applications and more accurate context included. 18 pages, 9 figures. v3: Minor edit prior to journal submission