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EVALUATIONS OF THE TWISTED ALEXANDER POLYNOMIALS OF 2-BRIDGE KNOTS AT ±1

MIKAMI HIRASAWA

Department of Mathematics, Nagoya Institute of Technology, Nagoya Aichi 466-8555, Japan

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and

KUNIO MURASUGI

Department of Mathematics, University of Toronto, Toronto, ON M5S2E4, Canada

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https://doi.org/10.1142/S0218216510008418Cited by:4 (Source: Crossref)

Abstract

Let H(p) be the set of 2-bridge knots K(r), 0<r<1, such that there is a meridian-preserving epimorphism from G(K(r)), the knot group, to G(K(1/p)) with p odd. Then there is an algebraic integer s₀ such that for any K(r) in H(p), G(K(r)) has a parabolic representation ρ into SL(2, ℤ[s₀]) ⊂SL(2, ℂ). Let be the twisted Alexander polynomial associated to ρ. Then we prove that for any K(r) in H(p), and , where , μ ∈ ℤ[s₀]. The number μ can be recursively evaluated.

Dedicated to Jose Maria Montesinos on the Occasion of his 65th Birthday

Keywords:

AMSC: 57M25, 57M27

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