Problem 1.8 — (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 $V_K(t)\in\mathbb Z[t,t^{-1}]$. Gromov--Witten invariants are suitably defined virtual counts of pseudoholomorphic curves in a symplectic target; open versions allow boundaries on specified Lagrangian submanifolds.

Statement recorded in the supplied source.

Develop the relationship between the Jones polynomial and Gromov--Witten invariants, including suitable curve-counting constructions and a precise identification of the knot information they recover. The source suggests the broader HOMFLY generating-function and Calabi--Yau/Lagrangian viewpoint as a route. Specify the target geometry and counting theory as part of the construction, rather than asserting a correspondence for an unspecified space. [S2]

Short English statement

Relate Jones and related knot polynomials to precise curve-counting theories in symplectic or Calabi--Yau geometry.

Sources

• S1 ULAM. AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400008 (AMR-103-0008). 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 1.8 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.