turkmath.org

Zariski-van Kampen theorem expresses the fundamental group of the complement of an algebraic curve on $C^2$ in terms of generators and monodromy relations. Therefore, the Alexander module of the curve is also (almost) expressed in terms of generators and monodromy relations. As far as the Alexander module of an n-gonal curve is concerned, the group of monodromy relations is a subgroup of the Burau group Bu_n, which is a certain subgroup of GL(n-1, Z[t,1/t]). For trigonal curves (n=3 case), Degtyarev gave a characterization of the monodromy groups: the monodromy group of a trigonal curve (except a trivial exceptional case) must be a finite index subgroup of $Bu_3$ whose image under the special epimorphism $Bu_3$ --> PSL(2,Z) is of genus 0 and conversely, most of such subgroups appear as monodromy groups of trigonal curves. However, this class of subgroups is still too large, hence it is not feasible to look at them all and determine their Alexander modules. In this talk, I plan to speak about a recently discovered method by which, given an abstract module over Z[t,1/t], one can determine whether or not it appears as the Alexander module of a trigonal curve. With this method, it should be feasible to determine all the Alexander modules.