On a Problem of Ferrand and Stoimenow Answered in Print by Rutherford and Ito: A Sharper Signature Bound for the HOMFLY Polynomial of Alternating Links, and a Short Proof that Quasipositive Alternating Links Are Positive

Abstract

Problem 1.16 of Ohtsuki's list (E. Ferrand, A. Stoimenow) asks whether every alternating link L satisfies σ(L) ≥ min deg_l P_L ≥ min deg_a F_L(a^-1, z), where σ is the signature, P_L(l, m) the HOMFLY polynomial and F_L(a, z) the Kauffman polynomial. Both inequalities are in print: the second is a corollary of Rutherford (2006), based on a proposition of Ng, and the first is a theorem of Ito (2026). This note does not solve the problem; it gives a short proof of the first inequality in a sharper form. For a connected reduced alternating diagram D of L with writhe w, n_- negative crossings, s Seifert circles and s_A circles in the all-A state, put d_-(D) = (n_- + s − s_A)/2. Then d_-(D) is a non-negative integer, positive if D has a negative crossing, and w − s + 1 ≤ min deg_l P_L ≤ σ(L) − d_-(D) = w − s + 1 + d_-(D). In particular σ(L) = min deg_l P_L if and only if D is positive. The proof compares degrees under the substitution which turns P into the Jones polynomial; its ingredients are classical (Morton's bound for the degree in m, the degree formula of Kauffman, Murasugi and Thistlethwaite, Traczyk's formula for the signature). The case of equality, and with it part (i) of a conjecture of Ito for alternating links and part (ii) for reduced alternating diagrams, also follows from Ito's theorem and an inequality in a paper of Stoimenow (2026), as we explain; the bound with d_-(D) is stronger when d_-(D) ≥ 3, and it gives a positive answer to a question of Orevkov (2020). With an elementary lemma on the signature and the nullity under a band move it yields a short proof that every reduced alternating diagram of a quasipositive alternating link is positive. This is the alternating case of the statement that homogeneous quasipositive links are positive, for which Stoimenow has announced a proof in a text on his home page; the question goes back to Baader (2005). We present a short independent proof of that case, not a new theorem. Exact computations, among them all prime alternating links with at most 14 crossings in all orientations and more than 50 000 quasipositive braid words, are tests of the statements; no proof depends on them. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: AMR-103-0016 (T. Ohtsuki (ed.), Problems on invariants of knots and 3-manifolds, Geom. Topol. Monogr. 4 (2002), Problem 1.16 of E. Ferrand and A. Stoimenow).