Tangles and generalizations of the Alexander polynomial
In the study of low-dimensional topology, one of the classical link invariants is the Multivariable Alexander Polynomial, or MVA, originally defined by Torres. More recently, Archibald constructed a generalization to an invariant of virtual tangles, i.e. tangles in thickened surfaces rather than 3-dimensional Euclidean space. This new invariant provides a straightforward verification of almost all the skein relations satisfied by the MVA. I will define a reduced version of Archibald’s invariant and discuss its relation to the Burau and Gassner representations for braids, as well as to Bar-Natan’s Alexander-type invariant of tangles without closed components.
Dr Iva Halacheva, University of Melbourne