Twistings and the Alexander polynomial

TL;DR

This paper provides an explicit formula for the Alexander polynomial of links formed by adding full twists to parallel strands, showing stabilization of the polynomial and utilizing quantum group representations for computation.

Contribution

It introduces a new explicit formula relating Alexander polynomials of twisted links to simpler cases, and demonstrates stabilization behavior after many twists.

Findings

Alexander polynomial formula for twisted links

Stability of Alexander polynomial after many twists

Use of quantum group representations in computation

Abstract

We give an explicit formula of the Alexander polynomial of the link obtained by adding an arbitrary number of full twists to positively oriented parallel n-strands in terms of the Alexander polynomials of the links obtained by adding 0,1,...,n-1 full twists. From this, we see that the Alexander polynomials stabilize after adding sufficiently many full twists. The main tool used in the computation is expressing the Alexander polynomial using the vector space representation of $U_{q}(gl(1|1))$.