Two strand twisting

TL;DR

This paper establishes limitations on unknotting fibred and two-bridge knots using specific two-strand twist moves, linking these constraints to knot invariants like the HOMFLY polynomial, crossing number, and braid index.

Contribution

It provides new bounds on the effectiveness of two-strand twist moves in unknotting knots, especially relating to fibred knots and two-bridge knots, based on knot invariants.

Findings

Fibred knots cannot be untied with ar{t}_{2k}-moves for all k  2.

An upper bound is given on the number of two-strand twists needed to unknot knots with non-trivial HOMFLY polynomial.

The braid index of two-bridge knots cannot be reduced by t_{2k}-moves for all but finitely many k.

Abstract

We prove that fibred knots cannot be untied with $\bar{t}_{2k}$-moves, for all $k \geq 2$. More generally, we give an upper bound on the number of two strand twist operations that allow to untie a knot with non-trivial HOMFLY polynomial, in terms of the minimal crossing number, and the braid index. As a by-product, we prove that the braid index of a two-bridge knot cannot be lowered by applying $t_{2k}$-moves, for all but finitely many $k \in \mathbb{N}$.