A Lagrangian representation of tangles II

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.

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Alexander Representation of Tangles

Bigelow

Cattabriga

Florens

2015

Acta Math Vietnam

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Coloured tangles and signatures

Cimasoni¹,

Conway²

2017

Math. Proc. Camb. Phil. Soc.

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Taking the signature of the closure of a braid defines a map from the braid group to the integers. In 2005, Gambaudo and Ghys expressed the homomorphism defect of this map in terms of the Meyer cocycle and the Burau representation. In the present paper, we simultaneously extend this result in two directions, considering the multivariable signature of the closure of a coloured tangle. The corresponding defect is expressed in terms of the Maslov index and of the Lagrangian functor defined by Turaev and the first-named author.

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A Lagrangian representation of tangles II

Abstract: Abstract. The present paper is a continuation of our previous paper [Topology 44 (2005), 747-767], where we extended the Burau representation to oriented tangles. We now study further properties of this construction.

Cited by 2 publications

References 12 publications

Alexander Representation of Tangles

Alexander Representation of Tangles

Coloured tangles and signatures

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