Alexander representation of tangles

TL;DR

This paper extends the Burau representation to tangles using Lescop's algebraic tools, providing an Alexander polynomial invariant for (1,1)-tangles and demonstrating its invariance through plat position techniques.

Contribution

It introduces a functor from the category of tangles to modules, generalizing the Alexander polynomial to tangles via an algebraic approach.

Findings

The invariant coincides with the Alexander polynomial for (1,1)-tangles.

A constructive proof of invariance using plat position is provided.

The extension applies to oriented tangles in the cylinder.

Abstract

A tangle is an oriented 1-submanifold of the cylinder whose endpoints lie on the two disks in the boundary of the cylinder. Using an algebraic tool developed by Lescop, we extend the Burau representation of braids to a functor from the category of oriented tangles to the category of Z[t,t^{-1}]-modules. For (1,1)-tangles (i.e., tangles with one endpoint on each disk) this invariant coincides with the Alexander polynomial of the link obtained by taking the closure of the tangle. We use the notion of plat position of a tangle to give a constructive proof of invariance in this case.