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Alexander Representation of Tangles
Stephen Bigelow
,
Alessia Cattabriga
,
Vincent Florens
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 ${\mathbb {Z}}[t,t^{-1}]$-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.