Problem 1.5 — (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.

For a knot $K$, let $M_n(K)$ be the $n$-fold cyclic cover of $S^3$ branched over $K$, obtained from the cyclic cover of its exterior determined by a meridian mapping to $1\in\mathbb Z/n\mathbb Z$ and then filling over the branch locus. Find a relationship between values of $V_K$ at roots of unity and the topology of the manifolds $M_n(K)$. Specify which values, cover invariants and relationships are valid, rather than assuming a formula for all $n$. [S2]

Short English statement

Relate root-of-unity values of the Jones polynomial to cyclic branched covers of the knot.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400005 (AMR-103-0005). 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 1.5 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.