Problem 1.13 — (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 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 $V_K(t)\in\mathbb Z[t,t^{-1}]$. The HOMFLY (skein) polynomial $P_L(l,m)$ is normalized by $P_O=1$ and $l^{-1}P_{L_+}-lP_{L_-}=mP_{L_0}$, where $L_+,L_-,L_0$ agree outside a ball and inside it have a positive crossing, a negative crossing and the oriented smoothing, respectively. 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 $\operatorname{span}_x=\deg_x-\min\deg_x$.

Statement recorded in the supplied source.

For each prescribed span $s$, is the set of distinct Jones polynomials of links with $\operatorname{span}_tV_L=s$ finite? Ask the same for $Q_L$ with fixed $\operatorname{span}_zQ_L$. For HOMFLY and Kauffman polynomials, ask finiteness of the set of values when the spans in both variables are fixed. The question counts distinct polynomial values, not link isotopy classes, and includes the knot cases. [S2]

Short English statement

Do fixed spans allow only finitely many values of the Jones, Q, HOMFLY and Kauffman polynomials, fixing both variable spans in the two-variable cases?

Sources

• S1 ULAM. AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400013 (AMR-103-0013). 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 1.13 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.