Braids, homomorphism defects of link invariants and combinatorial aspects of strongly invertible links
Settore scientifico disciplinare
MAT/03
Corso di studi
MATEMATICA
Relatori
tutor Prof. Lisca, Paolo
Parole chiave
Alexander theorem
braids
homomorphism defect
Markov theorem
Meyer cocycle
rho invariant
strongly invertible knots
strongly involutive links
twisted homology
twisted signature
Data inizio appello
22/03/2024
Consultabilità
Non consultabile
Data di rilascio
22/03/2027
Riassunto
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.