Problem 1.4 — (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.

Give an intrinsic explanation, from three-dimensional topology and the perturbative or configuration-space construction, of why the formal series obtained for $V_K(e^h)$ resums to a Laurent polynomial in $e^h$ with integer coefficients. Explain the special role, if any, of $S^3$. The sought explanation should not rely on first choosing a planar diagram or braid presentation, whose skein construction already proves polynomiality. [S2]

Short English statement

Explain intrinsically why the perturbative Jones series is an integer Laurent polynomial, without reducing the explanation to a diagram or braid formula.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400004 (AMR-103-0004). 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 1.4 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.