Chiral smoothings of knots

TL;DR

This paper investigates the conditions under which knots can be transformed into other knots via a single crossing smoothing, extending previous results to all torus knots T(2,m) except T(2,9), using Casson-Gordon and Heegaard Floer theories.

Contribution

It extends the classification of torus knots transformable into their mirror images by a single smoothing, applying Casson-Gordon theory to all T(2,m) except T(2,9).

Findings

T(2,5) can be converted into T(2,-5) with one smoothing.

T(2,5) is unique among T(2,m) for single smoothing to mirror image, assuming m is square free.

Casson-Gordon theory extends results to all T(2,m) except T(2,9).

Abstract

Given knots K and J, one can ask whether a single smoothing of a crossing in a diagram for K can convert it into a diagram for J. As an interesting example, Zekovic discovered that the torus knot T(2,5) can be converted into T(2,-5) with a single smoothing. On the other hand, Moore and Vasquez have shown that among torus knots of the form T(2,m), T(2,5) is the only one which can be converted into its mirror image with a single smoothing, assuming that m is square free. In this paper, Casson-Gordon theory is applied to extend the Moore-Vasquez result to all torus knots T(2,m), with the exception of T(2,9), which remains an open case. Applications of similar techniques to the general problem are also described, as are tools arising from Heegaard Floer theory.