The homomorphism defect of an extended Levine-Tristram signature via twisted homology

TL;DR

This paper extends the understanding of the homomorphism defect of Levine-Tristram signatures for braids and tangles by using a 4-dimensional perspective, connecting it to the Maslov index and generalizing previous formulas.

Contribution

It introduces a new 4-dimensional approach to evaluate the additivity defect, generalizing Gambaudo and Ghys' formula to colored braids and tangles.

Findings

Evaluation of the additivity defect via the Maslov index

Reformulation of the defect in terms of the Meyer cocycle and Gassner representation

Generalization of previous signature defect formulas to colored tangles

Abstract

Taking the Levine-Tristram signature of the closure of a braid defines a map from the braid group to the integers. A formula of Gambaudo and Ghys provides an evaluation of the homomorphism defect of this map in terms of the Burau representation and the Meyer cocycle. In 2017 Cimasoni and Conway generalized this formula to the multivariable signature of the closure of coloured tangles. In the present paper, we extend even further their result by using a different 4-dimensional interpretation of the signature. We obtain an evaluation of the additivity defect in terms of the Maslov index and the isotropic functor $\mathscr{F}_\omega$. We also show that in the case of coloured braids this defect can be rewritten in terms of the Meyer cocycle and the coloured Gassner representation, making it a direct generalization of the formula of Gambaudo and Ghys.