2012/03/20 by Stephen Bigelow, Bigelow, Stephen, Alessia Cattabriga +3 · 1 citation
Mathematics · #57M25 #57M27 #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:57M25 #msc:57M27
paper · pdf · doi:10.48550/arxiv.1203.4590
13 pages, 5 figures
openalex publication_date 2012/03/20 · arxiv created 2014/07/26 · arxiv updated 2014/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.