Problem 1.9 — (X.-S.

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 Jones polynomial is normalized by $V_O(t)=1$ and \[t^{-1}V_{L_+}(t)-tV_{L_-}(t)=(t^{1/2}-t^{-1/2})V_{L_0}(t),\] where the three oriented links agree outside a ball and inside it have respectively a positive crossing, a negative crossing, and the oriented smoothing. For knots $V_K(t)\in\mathbb Z[t,t^{-1}]$. A knot is alternating if it has a diagram whose successive crossings alternate between overcrossings and undercrossings along the knot. Since $V_K$ is Laurent, consider roots in $\mathbb C^*=\mathbb C\setminus\{0\}$.

Statement recorded in the supplied source.

Describe both sets \[\mathcal Z=\bigcup_{K}\{z\in\mathbb C^*:V_K(z)=0\},\qquad \mathcal Z_{\mathrm{alt}}=\bigcup_{K\ \mathrm{alternating}}\{z\in\mathbb C^*:V_K(z)=0\},\] including their distribution and limiting structure as the knots vary. Study also the suggested comparison, at bounded degree and crossing number, with zeros of integer polynomials whose coefficients sum to $1$. [S2]

Short English statement

Describe the complex zeros arising from Jones polynomials of all knots and of alternating knots, including their distribution and comparison with bounded-degree integer polynomials.

Sources

• S1 ULAM. AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400009 (AMR-103-0009). 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 1.9 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.