Problem 1.17 — (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 Conway polynomial $\nabla_L(z)$ is normalized by $\nabla_O=1$ and $\nabla_{L_+}-\nabla_{L_-}=z\nabla_{L_0}$ for the oriented skein triple. Write $\nabla_k(L)=[z^k]\nabla_L(z)$. The crossing number $c(L)$ is the least crossing count of a link diagram.

Statement recorded in the supplied source.

For every oriented link $L$ and integer $k\ge0$, is \[|\nabla_k(L)|\le \frac{c(L)^k}{2^k k!}?\] For $k=0$ the right side is $1$, including when $c(L)=0$. Investigate the source's refinements: whether the bound follows from Kontsevich--Drinfel'd methods and the Conway weight systems, whether parallel-oriented $(2,n)$ torus links attain the largest coefficients, and the corresponding coefficient bounds for other finite-type families such as Jones derivatives at $t=1$. [S2]

Short English statement

Bound every Conway coefficient by the proposed crossing-number expression, and investigate the extremal links and finite-type extensions suggested in the source.

Sources

• S1 ULAM. AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400017 (AMR-103-0017). 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 1.17 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.