Twisted Alexander polynomials of 2-bridge knots associated to metabelian representations

Details Twisted Alexander polynomials of 2-bridge knots associated to metabelian representations Twisted Alexander polynomials of 2-bridge knots associated to metabelian representations

2009-03-10

arxiv.org/abs/0903.1689v1

Mathematics

Geometric Topology

Scientific paper

Suppose the knot group G(K) of a knot K has a non-abelian representation \rho on A_4 \subset GL(4,Z). We conjecture that the twisted Alexander polynomial of K associated to \rho is of the form: \Delta_K(t)/(1-t) \phi(t^3), where \Delta_K (t) is the Alexander polynomial of K and \phi(t^3) is an integer polynomial in t^3. We prove the conjecture for 2-bridge knots K whose group G(K) can be mapped onto a free product Z/2*Z/3. Later, we discuss more general metabelian representations of the knot groups and propose a similar conjecture on the form of the twisted Alexander polynomials.

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