Tangles and Alexander type invariants

Abstract:

The Alexander polynomial is a classical invariant of knots and links in low dimensional topology, originally introduced by James Alexander and later extended to a multivariable invariant, MVA, by Torres. We study a generalization of the MVA by Archibald to virtual tangles (i.e. tangles in thickened surfaces) in the context of circuit algebras, construct a reduced version and relate it to an invariant defined by Bar-Natan in the context of metamonoids. It reduces to the Burau and Gassner representations on braids and admits a partially defined trace operation.