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

2009/03/01 by Mikami Hirasawa, Hirasawa, Mikami, Kunio Murasugi +1 · 1 citation
Mathematics · #57M25 #57M27 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.0903.0147

openalex publication_date 2009/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be an odd prime and D_p a dihedral group of order 2p. Let ρ: 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 Δ_ρ,K (t) be the twisted Alexander polynomial of K associated to ρ. 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), Δ_ρ,K(t) is of the form Δ_K(t)/(1-t) f(t) f(-t) for some integer polynomial f(t), where Δ_K (t) is the Alexander polynomial of K. Further, it is proved that f(t) ≡ Δ_K (t)/(1+t)ⁿ (mod p). Later we discuss the twisted Alexander polynomial associated to the general metacyclic representation.

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• Higher dimensional twisted Alexander polynomials for metabelian representations

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