Kirby Problem 1.7

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

All knots and links below are smooth in the oriented $3$-sphere, up to ambient isotopy, unless another ambient manifold is specified. Normalize the one-variable Alexander polynomial symmetrically, allowing multiplication by $\pm t^k$, and write its consecutive coefficients as $a_0,\ldots,a_n$ after removing zero coefficients outside its support. Set $b_i=|a_i|$; if the polynomial vanishes, the corresponding coefficient assertions are vacuous. A link is alternating if it has an alternating over/under diagram, and positive if it has an oriented diagram with every crossing positive. Its signature is that of the symmetrized Seifert form. For a $c$-component positive link, write the normalized Conway polynomial, defined by $\nabla_{L_+}-\nabla_{L_-}=z\nabla_{L_0}$ and $\nabla_U=1$, as $z^{c-1}\sum_i d_i z^{2i}$.

Statement recorded in the supplied source.

(a) For alternating knots, is $b_0,\ldots,b_n$ trapezoidal: nondecreasing up to the center, with any equality before the center extending to a central plateau, followed symmetrically by nonincrease? (b) For alternating links, is $b_i^2\geq b_{i-1}b_{i+1}$ for every internal $i$, with equality before the center only when $a_{i-1}=-a_i=a_{i+1}$, and with the number of consecutive equalities on the top plateau at most $|\sigma(L)|$? (c) For positive links, is the sequence $(d_i)$ strictly log-concave, $d_i^2>d_{i-1}d_{i+1}$ at every internal index? The alternating restriction in (a), and the meaning of "concave" as a trapezoidal sequence, are specified by the original remarks. [S1][S2]

Short English statement

Ask for trapezoidal Alexander coefficients of alternating knots, a stronger log-concavity and signature bound for alternating links, and strict log-concavity of Conway coefficients for positive links.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2666 (KP-1.7). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, author preliminary edition (2026), Problem 1.7 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.