Enrico Manfredi (Bologna): Computation of Alexander polynomials of links in lens spaces from different representations

 
Abstract: Links in lens spaces can be represented by different types of diagrams, including disk, mixed link, punctured disk and grid diagrams. For each of them it is possible to compute a presentation for the fundamental group of the complement of the link. Moreover, from the group presentation one can determine the Alexander polynomial and a wider class of link invariants, called twisted Alexander polynomials. The first three representations lead to distinct generalizations of the Wirtinger theorem, and we will show how to keep track of the torsion in the group using a particular class of Alexander polynomials twisted by one-dimensional representations. Furthermore, when the Alexander polynomial of a knot in the 3-sphere is computed from the grid diagram one can see a connection to the Knot Floer Homology, and we will show a presentation of the group that can lead to a similar relation for knots in lens spaces.