ON THE TWISTED ALEXANDER POLYNOMIAL FOR REPRESENTATIONS INTO SL2(ℂ)

Abstract

We study the twisted Alexander polynomial ΔK,ρof a knot K associated to a non-abelian representation ρ of the knot group into SL2(ℂ). It is known for every knot K that if K is fibered, then for every non-abelian representation, ΔK,ρis monic and has degree 4g(K) – 2 where g(K) is the genus of K. Kim and Morifuji recently proved the converse for 2-bridge knots. In fact they proved a stronger result: if a 2-bridge knot K is non-fibered, then all but finitely many non-abelian representations on some component have ΔK,ρnon-monic and degree 4g(K) – 2. In this paper, we consider two special families of non-fibered 2-bridge knots including twist knots. For these families, we calculate the number of non-abelian representations where ΔK,ρis monic and calculate the number of non-abelian representations where the degree of ΔK,ρis less than 4g(K) – 2.