On the twisted Alexander polynomial and the A-polynomial of 2-bridge knots

We show that the A-polynomial A(L,M) of a 2-bridge knot b(p, q) is irreducible if p is prime, and if (p − 1)/2 is also prime and q , 1 then the L-degree of A(L,M) is (p−1)/2. This shows that the AJ conjecture relating the A-polynomial and the colored Jones polynomial holds true for these knots, according to work of the second author. We also study relationships between the A-polynomial of a 2-bridge knot and a twisted Alexander polynomial associated with the adjoint representation of the fundamental group of the knot complement. We show that for twist knots the A-polynomial is a factor of the twisted Alexander polynomial. 1. Background and conventions 1.1. Representation variety. Let K be a knot in S 3 and X = S 3 \ K be its complement. Let π = π1(X) be the fundamental group of the complement. Let R(π) = Hom ( π,SL(2,C) ) be the set of representations of π to SL(2,C). This is a complex affine algebraic set, which is called the representation variety, although it might be a union of a finite number of (irreducible) algebraic varieties in the sense of algebraic geometry. The group SL(2,C) acts on R(π) by conjugation. The algebro-geometric quotient of R(π) under this action is called the character variety of π, denoted by X(π). The character of a representation ρ is the map χρ : π → C determined by χρ(γ) = tr ρ(γ), for γ ∈ π. There is a bijection between X(π) and the set of characters of representations of π. 1.2. The A-polynomial. Let B = (μ, λ) be a pair of meridian-longitude of the boundary torus of X. Let RU be the subset of R(π) containing all representations ρ such that ρ(μ) and ρ(λ) are upper triangular matrices: (1.1) ρ(μ) = M ∗ 0 M−1  , ρ(λ) = L ∗ 0 L−1  (any representation can be conjugated to have this form). Then RU is an algebraic set, because we only add the requirement that the lower left entries of ρ(μ) and ρ(λ) 2000 Mathematics Subject Classification. 57M27.