Braids, homomorphism defects of link invariants and combinatorial aspects of strongly invertible links

My Ph.D. thesis is divided in three main parts. In part one I focus on a generalization of a formula of Gambaudo and Ghys, studied by Cimasoni and Conway. Taking the Levine-Tristram signature of the closure of a braid defines a map from the braidgroup to the integers. A formula of Gambaudo and Ghys provides an evaluation of the homomorphism defect of this map in terms of two classic objects in low dimensional topology: 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. I extend even further their result by using a different 4-dimensional interpretation of the signature. I obtain an evaluation of the homomorphism defect in terms of two objects that generalize the Meyer cocycle and the Burau representation to the setting of coloured tangles: the Maslov index and the isotropic functor Fω. I also show that in the case of coloured braids this result is a direct generalization of the formula of Gambaudo and Ghys. In part two I focus on an invariant for closed 3-manifolds with an approach similar to part one. This invariant is called the Atiyah-Patodi-Singer rho-invariant and it generalizes the multivariate Levine-Tristram signature. This was used by Cochran, Harvey and Horn to define many real valued quasimorphisms on subgroups of the mapping class group of a compact surface with boundary. The homomorphism defect of these quasimorphisms, similarly to the Levine-Tristram signature case, turns out to be the difference of the twisted and untwisted signature of a 4-manifold, and I show that in some cases it can be evaluated purely in terms of the Meyer cocycle. Fixing some boundary conditions, the rho-invariant was also defined by Kirk and Lesch for 3-manifolds with non-empty boundary. When the boundary is toroidal these invariants were studied by Toffoli. These invariants allow us to define some real-valued maps on subgroups of the braid groups and I calculate the homomorphism defect of these maps in terms of the Meyer cocycle and a corrective term. As a special case, I get another proof of the multivariate Gambaudo–Ghys formula. The third part focuses on knots and links with a special symmetry, called strongly invertible.I first show a result on the equivariant tube equivalence for invariant Seifert surfaces of strongly invertible knots. This is joint work with Collari, Di Prisa and Framba. Finally, I study an equivariant version of Alexander and Markov theorems. A classic result by Alexander (1923) establishes that every link is the closure of a braid. Moreover Markov theorem (1936) characterizes the equivalence of braids yielding the same link closure through a set of moves: conjugation, stabilization, and destabilization. Extending this framework, I define an equivariant closure function that, given two palindromic braids (i.e. braids with a particular symmetry), yields a strongly involutive link. Strongly involutive links are a class of links that are preserved by an involution. This class contains strongly invertible knots and links. I prove an equivariant Alexander theorem, showing that every strongly involutive link is equivalent to the equivariant closure of two palindromic braids. Furthermore, I establish a set of equivariant moves that generalize the classic Markov moves, extending the theory to the equivariant setting.