Problem 1.18 — (A.

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.

A knot or link is a smooth oriented embedding of one or more circles in the oriented $3$-sphere, considered up to orientation-preserving ambient isotopy. Write $O$ for the unknot, which bounds an embedded disk. The Kauffman polynomial is $F_L(a,z)=a^{-w(D)}[D]$, where $w(D)$ is diagram writhe. The regular-isotopy bracket has $[O]=1$, $[D_+]+[D_-]=z([D_0]+[D_\infty])$, and a positive or negative curl multiplies it by $a$ or $a^{-1}$ respectively; the two smoothings in this relation are unoriented. Put $Q_L(z)=F_L(1,z)$. For a nonzero Laurent polynomial, $\deg_x$ and $\min\deg_x$ are the greatest and least exponents of $x$ with nonzero coefficient, and $\operatorname{span}_x=\deg_x-\min\deg_x$. Let $\chi(L)$ be the greatest Euler characteristic of an oriented Seifert surface for $L$ with no closed components. Let $u(K)$ be the fewest crossing changes needed to turn a knot $K$ into the unknot.

Statement recorded in the supplied source.

Do both inequalities \[\min\deg_aF_L(a^{-1},z)\le1-\chi(L)\quad\text{for every link }L,\] \[\min\deg_aF_K(a^{-1},z)\le2u(K)\quad\text{for every knot }K\] hold? The substitution $a\mapsto a^{-1}$ is made before taking the least $a$-exponent. Neither statement is restricted to alternating links. [S2]

Short English statement

Do the two proposed Kauffman minimal-degree inequalities hold against Seifert-surface Euler characteristic and knot unknotting number?

Sources

• S1 ULAM. AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400018 (AMR-103-0018). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 T. Ohtsuki (editor), Problems on invariants of knots and 3-manifolds, Geometry & Topology Monographs 4 (2002), pp. 377--572 (published 2004), item 1.18 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.