Torus knots, the A-polynomial, and SL(2,C)

The A-polynomial of a knot is defined in terms of SL(2,C) representations of the knot group, and encodes information about essential surfaces in the knot complement. In 2005, Dunfield-Garoufalidis and Boyer-Zhang proved that it detects the unknot using Kronheimer-Mrowka’s work on the Property P conjecture. Here we use more recent results from instanton Floer homology to prove that a version of the A-polynomial detects whether a knot is a torus knot. We moreover completely determine which individual torus knots are detected by this A-polynomial. These results enable progress towards a folklore conjecture about boundary slopes of non-torus knots. Finally, we use similar ideas to prove that a knot in the 3-sphere admits infinitely many SL(2,C)-abelian Dehn surgeries if and only if it is a torus knot, affirming a variant of a conjecture due to Sivek-Zentner.

💡 Research Summary

This paper establishes two powerful detection results for torus knots using the A‑polynomial and SL(2,ℂ) representation theory, building on recent advances in instanton Floer homology.

The first main theorem (Theorem 1.1) proves that a knot K ⊂ S³ admits infinitely many SL(2,ℂ)-abelian Dehn surgeries if and only if K is a torus knot. Here “SL(2,ℂ)-abelian” means that every homomorphism π₁(S³_r(K)) → SL(2,ℂ) has abelian image. The proof proceeds by showing that any knot with infinitely many such surgeries must be SU(2)-averse, which by the work of Sivek–Zentner forces it to be an instanton L‑space knot. Instanton L‑space knots are known to be fibered, strongly quasipositive, and to satisfy stringent constraints on the coefficients of their Alexander polynomial (they lie in {−1,0,1} and alternate in sign). Using these constraints together with satellite knot analysis, the authors rule out all non‑torus possibilities, concluding that only torus knots can have the required infinite family of SL(2,ℂ)-abelian surgeries.

The second main theorem (Theorem 1.2) concerns the “thin” condition on the enhanced A‑polynomial ˜A_K(M,L). The enhanced A‑polynomial discards the abelian factor L−1 and records only the irreducible SL(2,ℂ) representations. Its Newton polygon N(K) is a convex lattice polygon in ℝ²; K is called thin if N(K) lies on a single line. Dunfield–Garoufalidis and Boyer–Zhang previously showed that N(K) being a point characterizes the unknot. The authors extend this dramatically: a knot is thin if and only if it is a torus knot. The argument again uses instanton L‑space theory: thinness forces the knot to be an instanton L‑space knot, which together with the Alexander polynomial coefficient restrictions eliminates all non‑torus possibilities, including hyperbolic and satellite knots.

Corollary 1.3 refines the detection power of ˜A_K: it distinguishes a torus knot T_{a,b} precisely when |a|=2 or |b|=2, or when |a| and |b| are powers of distinct primes. For example, ˜A_K = ˜A_{T_{3,25}} uniquely identifies T_{3,25}, whereas ˜A_K = ˜A_{T_{3,35}} only tells us that K is some torus knot among T_{3,35}, T_{5,21}, T_{7,15}. Corollary 1.4 shows that the pair (˜A_K, deg Δ_K(t)) detects every torus knot, improving earlier results that required knot Floer homology.

The paper also explores implications for boundary slopes. Since each side of the Newton polygon of the original A‑polynomial yields a boundary slope, a torus knot has exactly two slopes (the cabling slope ab and the Seifert slope 0). The authors prove a partial converse (Theorem 1.7): a non‑torus fibered knot must either admit a separating incompressible surface meeting the boundary torus in slope 0 or have at least three distinct boundary slopes. This provides progress toward the folklore conjecture that only torus knots have exactly two boundary slopes.

The structure of the paper is as follows. Section 2 reviews instanton L‑spaces, SU(2)-averse knots, and the A‑polynomial, establishing key lemmas about Alexander polynomial coefficients and satellite patterns. Section 3 develops new constraints on satellite instanton L‑space knots, showing that both pattern and companion must be fibered and nontrivial. Section 4 applies these results to prove Theorem 1.2, Corollaries 1.3 and 1.4, and the boundary‑slope result Theorem 1.7. Section 5 uses the SU(2)-averse classification together with instanton techniques to prove Theorem 1.1, completing the characterization of knots with infinitely many SL(2,ℂ)-abelian surgeries.

Overall, the work demonstrates that the enhanced A‑polynomial, together with elementary invariants such as the Alexander polynomial degree, provides a complete algebraic fingerprint for torus knots, and that instanton Floer homology supplies the analytic machinery needed to translate this fingerprint into topological detection results. The methods open avenues for extending these detection theorems to knots in other 3‑manifolds and for further exploring the relationship between representation varieties, boundary slopes, and Floer‑theoretic invariants.