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- Title
ON THE TWISTED ALEXANDER POLYNOMIAL FOR REPRESENTATIONS INTO SL<sub>2</sub>(ℂ).
- Authors
TRAN, ANH T.
- Abstract
We study the twisted Alexander polynomial ΔK,ρ of 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,ρ is 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,ρ non-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,ρ is monic and calculate the number of non-abelian representations where the degree of ΔK,ρ is less than 4g(K) - 2.
- Subjects
POLYNOMIALS; REPRESENTATION theory; KNOT theory; NONABELIAN groups; PROOF theory; NUMBER theory
- Publication
Journal of Knot Theory & Its Ramifications, 2013, Vol 22, Issue 10, p-1
- ISSN
0218-2165
- Publication type
Article
- DOI
10.1142/S0218216513500594