Kirby Problem 1.29

Definitions and mathematical statement

A knot $K\subset S^3$ is hyperbolic if its complement admits a complete finite-volume metric of constant sectional curvature $-1$. Write $\operatorname{vol}(K)$ for that metric's volume. Define $\det(K)=|\Delta_K(-1)|$, where $\Delta_K$ is its one-variable Alexander polynomial normalized by $\Delta_K(1)=1$ and symmetry. Let $\widetilde{\mathrm{Kh}}^{i,j}(K;\mathbb Q)$ be reduced Khovanov homology over $\mathbb Q$, with its homological and quantum gradings. Set \[ R(K)=\sum_{i,j}\dim_{\mathbb Q}\widetilde{\mathrm{Kh}}^{i,j}(K;\mathbb Q). \] Equivalently this is the total free rank of reduced integral Khovanov homology, so torsion contributes nothing to $R(K)$.

The questions are whether the following strict bounds hold:

1. $\operatorname{vol}(K)<2\pi\log\det(K)$ for every alternating hyperbolic knot;
2. $\operatorname{vol}(K)<2\pi\log R(K)$ for every hyperbolic knot.

Here alternating means admitting a diagram with alternating overpasses and underpasses, and $\log$ is the natural logarithm. These compare the volume of the complete knot complement with discrete invariants of that same knot; no choice of a particular diagram occurs in either inequality.

Short English statement

Can determinant, and more generally reduced Khovanov rank, always bound a hyperbolic knot's volume by these logarithmic formulas?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2688 (KP-1.29), 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 1.29. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.