Kirby Problem 1.26

Definitions and mathematical statement

Let $K\subset S^3$ be a knot, and let $J_n(K;q)$ denote its zero-framed $\mathfrak{sl}_2$ colored Jones polynomial in the standard variable $q$, colored by the $n$-dimensional representation. Use the unreduced normalization $J_n(U;q)=(q^{n/2}-q^{-n/2})/(q^{1/2}-q^{-1/2})$. Let $d_+(n)$ and $d_-(n)$ be the largest and smallest powers of $q$ occurring. The set of Jones slopes is \[ js(K)=\operatorname{Clust}_{n\to\infty}\frac{4d_+(n)}{n^2} \ \cup\ \operatorname{Clust}_{n\to\infty}\frac{4d_-(n)}{n^2}, \] where cluster points are limits along infinite subsequences.

A boundary slope $p/q$ is realized by a properly embedded orientable surface in $X=S^3\setminus\operatorname{int}\nu K$, with nonempty essential boundary curves of class $p\mu+q\lambda$, that is incompressible, boundary-incompressible, and not boundary parallel. Incompressibility rules out compressing disks; boundary-incompressibility rules out disks giving compressions along essential arcs on the surface.

The Slope Conjecture asks whether every member of $js(K)$ is a boundary slope of such a surface, for every knot $K$. The inclusion in the boundary-slope set is the conjectural target supplied by the source notes after their definition of $js(K)$.

Short English statement

Must every limiting quadratic degree of the colored Jones polynomials arise from an essential surface in the knot exterior?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2685 (KP-1.26), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.26. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.