Alexander representation of tangles

Details Alexander representation of tangles Alexander representation of tangles

2012-03-20

arxiv.org/abs/1203.4590v1

Mathematics

Geometric Topology

16 pages, 5 figures

Scientific paper

A tangle is an oriented 1 -submanifold of the cylinder whose endpoints lie on the two disks in the boundary of the cylinder. We define an invariant of tangles using an algebraic tool developed by Lescop. Introducing the notion of plat position of a tangle, we give a constructive proof of its invariance. Moreover, we prove that for (1, 1) -tangles (i.e., tangles with one endpoint on each disk) the invariant coincides with the Alexander polynomial of the link obtained taking the closure of the tangle. At the end of the paper we analyze a functorial approach to the invariant, describing it as a particular case of a more general construction developed in a forthcoming paper by Florens and Massuyeau

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