Problem 1.16 — (E.

Definitions and mathematical statement

An alternating link admits a diagram in which successive crossings encountered along each component alternate over and under. Let $\sigma(L)$ be its signature, $P_L(l,m)$ its HOMFLYPT polynomial, and $F_L(a,z)$ its Kauffman polynomial, all with the source's intended orientation and normalization conventions.

For a nonzero Laurent polynomial $Q(t,u)=\sum_j t^j q_j(u)$, define $\min\deg_t Q=\min\{j:q_j\ne0\}$. Reading the flattened "$a-1$" argument as $a^{-1}$, the question attributed to E. Ferrand and A. Stoimenow is whether every alternating link satisfies \[ \sigma(L)\geq\min\deg_l P_L(l,m) \geq\min\deg_a F_L(a^{-1},z). \] Under this reciprocal reading, the last degree is $-\max\deg_a F_L(a,z)$.

The record does not provide the skein/normalization conventions or the sign of signature. A usual signature convention is the signature of $V+V^T$ for a Seifert matrix $V$, but its compatibility with the displayed polynomial conventions must be checked. Variable inversion and signature-sign reversal change the stated inequalities. The reciprocal interpretation of "$a-1$" is explicit rather than silently replacing it by either a reciprocal or an additive substitution. No knot-only, prime, or nonsplit restriction is added.

Short English statement

For every alternating link, compare signature with the minimum HOMFLYPT degree and the minimum degree of the reciprocal-variable Kauffman polynomial. Fix signs, skein normalizations, and the flattened reciprocal notation.

Sources

• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 1.16, PDF page 20, attributed to E. Ferrand and A. Stoimenow, https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf; accessed 2026-07-29, PDF-font-structure extraction.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400016$, source code catalogue entry, Problem 1.16. Dataset attribution: Ulam AI and the UnsolvedMath Contributors. The source title field is truncated, but both proposers appear in the supplied statement.