A note on the X-torsion order of a knot

TL;DR

This paper introduces a method to compute the X-torsion order of a knot using a spectral sequence, enabling extensive calculations and an example with order 4.

Contribution

It provides a novel computational approach for the X-torsion order of knots based on the Lee complex and spectral sequences.

Findings

Calculated X-torsion order for various knots

Provided an explicit example with order 4

Demonstrated the effectiveness of spectral sequence methods

Abstract

We show that the $X$-torsion order of a knot, which is defined in terms of a generalised Lee complex, can be calculated using the reduced Bar-Natan--Lee--Turner spectral sequence. We use this for extensive calculations, including an example of $X$-torsion order $4$.