Every nontrivial knot in $S^3$ has nontrivial A-polynomial

Abstract

We show that every nontrivial knot in the 3-sphere has a nontrivial A-polynomial. In Theorem 1 of [3], Kronheimer and Mrowka give a proof of the following remarkable theorem, thereby establishing the truth of the Property P conjecture. Theorem 0.1 (Kronheimer-Mrowka). Let K be any nontrivial knot in S and let M(r) be the manifold obtained by Dehn surgery on K with slope r with respect to the standard meridian-longitude coordinates of K. If |r| ≤ 2, then there is an irreducible homomorphism from π1(M(r)) to SU(2). The purpose of this note is to describe another consequence of Theorem 0.1, answering a question which has been around for about ten years. We show Theorem 0.2. Every nontrivial knot K in S has nontrivial A-polynomial. The A-polynomial was introduced in [1]. We recall its definition for a knot in S. For a compact manifold W , we use R(W ) and X(W ) to denote the SL2(C) representation variety and character variety of W respectively, and q : R(W ) → X(W ) to denote the quotient map sending a representation ρ to its character χρ (see [2] for detailed definitions). Note that q is a regular map between the two varieties defined over the rationals. LetK be a knot in S,M its exterior, and {μ, λ} the standard meridian-longitude basis for π1(∂M). Let i∗ : X(M) → X(∂M) be the restriction map, also regular, induced by the homomorphism i∗ : π1(∂M) → π1(M), and let Λ be the set of diagonal representations of π1(∂M), i.e. Λ = {ρ ∈ R(∂M) | ρ(μ), ρ(λ) are both diagonal matrices}. Then Λ is a subvariety of R(∂M) and q|Λ : Λ → X(∂M) is a degree 2, surjective, regular map. We may identify Λ with C∗ × C∗ through the eigenvalue map E : Λ → C∗ × C∗ which sends ρ ∈ Λ to (x, y) ∈ C∗ × C∗ if ρ(μ) = ( x 0 0 x−1 ) and ρ(λ) = ( y 0 0 y−1 ). Let Received by the editors May 4, 2004 and, in revised form, May 12, 2004. 2000 Mathematics Subject Classification. Primary 57M27, 57M25; Secondary 57M05.