Problem 1.6 — (J.

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 $V_K(t)\in\mathbb Z[t,t^{-1}]$.

Statement recorded in the supplied source.

Find a geometric relationship between Jones polynomials and counts of points of algebraic varieties over finite fields. For a variety $X$ defined over $\mathbb F_q$, such a count means $\#X(\mathbb F_q)$. Determine a meaningful construction of varieties or geometric counting expressions associated to a knot that explains its Jones polynomial. Allow appropriate signed or cohomological expressions when needed; the question does not prescribe one universally positive point-count formula. [S2]

Short English statement

Find an algebraic-geometric interpretation of Jones polynomials through finite-field point counts or their appropriate geometric refinements.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400006 (AMR-103-0006). 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 1.6 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.