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

Abstract

Let p be an odd prime and D_p a dihedral group of order 2p. Let \rho: G(K) --> D_p --> GL(p,Z) be a non-abelian representation of the knot group G(K) of a knot K in 3-sphere. Let \Delta_{\rho,K} (t) be the twisted Alexander polynomial of K associated to \rho. Let H(p) is the set of 2-bridge knots K, such that G(K) is mapped onto a non-trivial free product Z/2 * Z/p. Then we prove that for any 2-bridge knot K in H(p), \Delta_{\rho,K}(t) is of the form \Delta_{K}(t)/(1-t) f(t) f(-t) for some integer polynomial f(t), where \Delta_K (t) is the Alexander polynomial of K. Further, it is proved that f(t) \equiv {\Delta_K (t)/(1+t)}^n (mod p). Later we discuss the twisted Alexander polynomial associated to the general metacyclic representation.