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

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 s0such 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 [Formula: see text] be the twisted Alexander polynomial associated to ρ. Then we prove that for any K(r) in H(p), [Formula: see text] and [Formula: see text], where [Formula: see text], μ ∈ ℤ[s0]. The number μ can be recursively evaluated.