The leading coefficient of the Alexander polynomial of twisted torus knots

Abstract

For a twisted torus knot K(p,q;r,s) with gcd(p,q) = 1 and 2 ≤ r < p we determine the leading coefficient of the Alexander polynomial Δ exactly. Writing Δ up to a unit as Δ̃(u) = Σ_{m=0}^{r−1} γ_m(t) u^m with u = t^{rs}, we prove a multiset law expressing each γ_m as a sum of p monomials with all coefficients +1, a closed formula for the degree sequence d_m = deg γ_m in terms of return times of a rotation, and the resulting leading-coefficient law |lc(Δ)| = #{m : d_m + mrs maximal}. Consequently Δ is monic if and only if that maximum is attained once, and Δ fails to be monic exactly when −rs is an edge slope of the upper convex hull of {(m, d_m)}; in particular at most r−1 values of s are non-monic for fixed (p,q,r), however large |s|. We then classify the extremal case |lc(Δ)| = r completely, by three explicit arithmetic conditions on (p,q,r), for every admissible parameter. This supplies the algebraic half of a conjecture of Adnan and Park; we do not address its geometric direction. Prior work. Two papers by Sangyop Lee (J. Knot Theory Ramifications 2024 and 2026) concern the same family at r = 2: the first determines which of these knots are cable or torus knots, the second determines their knot types outright. We have not been able to consult either paper directly and rely on their abstracts. Since a knot type determines its Alexander polynomial, the case r = 2 of the leading-coefficient law may well be deducible from Lee (2026), and we claim no priority there. For r ≥ 3 we are not aware of a classification from which our statements would follow. The case q | r (Δ always monic) is attributed to Lee (2012). This work was developed in collaboration with the AI system Claude Code (Anthropic): the geometric family, the guiding questions, and the computing hardware are due to the author (a sculptor); the formalisation, the algorithms and the proofs were developed jointly with Claude Code (Anthropic). The collaboration is stated as a fact, not a disclaimer. The author takes full responsibility for all contents. Website: https://knot-structures.stainlesssteel4u.de/