Kirby Problem 2.13

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 $S$ be an oriented finite-type surface and $\Gamma\subset S$ a finite embedded graph, considered up to ambient isotopy. A graph surgery is permitted when a component $D$ of $S\setminus\Gamma$ is a disk. Choose oriented edges $x,y$ on its boundary and an arc $\lambda$ in $D$ between their interiors. A small regular neighborhood $N$ of $\lambda$ is a rectangle with opposite sides lying in $x,y$. Remove those two subarcs from the graph and replace them by the other two sides of $N$, which run parallel to $\lambda$. This is the source's surgery move. Intermediate complements need not remain disks, and no trivalence, four-valence or unicellular hypothesis is imposed.

Statement recorded in the supplied source.

Give necessary and sufficient conditions on two ambient-isotopy classes of embedded finite graphs $\Gamma_1,\Gamma_2\subset S$ for a finite sequence of these surgeries, with isotopies between steps, to transform one into the other. [S2]

Short English statement

Characterize exactly when two embedded graphs on a surface are connected by the source’s disk-face surgery moves, without restricting to unicellular or fixed-valence cases.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2761 (KP-2.13). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, eds., K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 2.13 and its remarks; Chapters 1 and 2 for the stated conventions. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.