Kirby Problem 1.23

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.

Let $K$ be a finite trivalent graph embedded in $\mathbb R^2$. A Tait coloring labels each edge by one of three specified colors, with three distinct colors at every vertex; $\mathrm{Tait}(K)$ counts these labelings. Write $J^{\sharp}(K)$ for the finite-dimensional $\mathbb F_2$ instanton vector space of Kronheimer--Mrowka, and $J^{\flat}(K)$ for its planar combinatorial foam counterpart, as constructed by Khovanov--Robert; these are the sharp and flat superscripts in the original source. An embedded bridge is an edge separated from the remainder by an embedded sphere intersecting that edge once.

Statement recorded in the supplied source.

(a) Is $\dim_{\mathbb F_2}J^{\sharp}(K)=\mathrm{Tait}(K)$ for every such $K$? (b) If $K$ is bridgeless, is $J^{\flat}(K)\ne0$? [S1][S2]

Short English statement

Determine whether instanton web homology counts three-edge colorings, and whether the planar foam counterpart is nonzero for every bridgeless planar trivalent graph.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2682 (KP-1.23). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, author preliminary edition (2026), Problem 1.23 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.