Alexander-Lin twisted polynomials

TL;DR

This paper generalizes the twisted Alexander polynomial to arbitrary finitely presented groups, extends fibering obstruction theorems, and explores virtual knots with trivial Jones polynomials, revealing new insights into knot invariants.

Contribution

It introduces a broader definition of twisted Alexander polynomials and extends fibering obstruction results to more general groups.

Findings

Generalized twisted Alexander polynomial for arbitrary finitely presented groups

Extended fibering obstruction theorem to new group classes

Identified virtual knots with trivial Jones polynomial but nontrivial invariants

Abstract

X.S. Lin's original definition of twisted Alexander knot polynomial is generalized for arbitrary finitely presented groups. J. Cha's fibering obstruction theorem is generalized. The group of a nontrivial virtual knot shown by L. Kauffman to have trivial Jones polynomial is seen also to have a faithful representation that yields a trivial twisted Alexander polynomial.