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

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 colored tangles. In this 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 [Formula: see text]. We also show that in the case of colored braids this defect can be rewritten in terms of the Meyer cocycle and the colored Gassner representation, making it a direct generalization of the formula of Gambaudo and Ghys.