2-strand twisting and knots with identical quantum knot homologies

TL;DR

This paper investigates how twisting strands in a knot affects its Khovanov and Khovanov-Rozansky homologies, providing new methods to generate knots with identical homologies and identifying mutant pairs with isomorphic reduced homologies.

Contribution

It introduces long exact sequences and algebraic structures to analyze homology changes under strand twists, enabling new knot generation and mutant pair identification.

Findings

Generated infinite families of knots with identical homologies

Identified mutant pairs with isomorphic reduced homologies

Developed algebraic tools for homology analysis

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.