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

Problem 1.16 of Ohtsuki's list (E. Ferrand, A. Stoimenow) asks whether every alternating link \(L\) satisfies \(\sigma(L)\)\({}\ge\operatorname{min\,deg}_l P_L\)\({}\ge\operatorname{min\,deg}_a F_L(a^{-1},z)\), where \(\sigma\) 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 \(d_-(D)\)\({}=\frac12(n_-\)\({}+s\)\({}-s_A)\). Then \(d_-(D)\) is a non-negative integer, positive if \(D\) has a negative crossing, and \(w\)\({}-s\)\({}+1\ \)\({}\le\ \operatorname{min\,deg}_l P_L\ \)\({}\le\ \sigma(L)\)\({}-d_-(D)\ \)\({}=\ w\)\({}-s\)\({}+1\)\({}+d_-(D).\) In particular \(\sigma(L)\)\({}=\operatorname{min\,deg}_l P_L\) 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 \(d_-(D)\ge3\), 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.