Kirby Problem 3.32

Definitions and mathematical statement

Give a tame closed embedded surface in $\mathbb R^3$ by a finite triangulated polyhedral surface whose vertices have rational coordinates, with faces meeting only along their common subfaces. Smooth surfaces may instead be represented by equivalent finite triangulation data. Wild embeddings and arbitrary unencoded continuous maps are not inputs to the algorithmic problem.

Two such surfaces $S_0,S_1$ are ambiently isotopic if there is a continuous family of homeomorphisms $H_t:\mathbb R^3\to\mathbb R^3$, $0\leq t\leq1$, with $H_0=\operatorname{id}$ and $H_1(S_0)=S_1$. Work in the corresponding tame PL or smooth category. The surfaces are compared as embedded subsets, without a prescribed parametrization of their points.

Is there an algorithm that takes two finite descriptions and decides whether the surfaces are ambiently isotopic, halting in every case? The source's principal case is that both surfaces are connected, with arbitrary genus. Disconnected inputs additionally involve their relative placement.

Matching abstract surface genus is insufficient, since the embedding can be knotted. Even homeomorphism information for complementary regions must be made compatible along their boundary surfaces to yield the required ambient isotopy. The requested conclusion is decidability, without a polynomial-time requirement.

Short English statement

Is ambient isotopy of closed tame surfaces in three-dimensional Euclidean space algorithmically decidable from finite descriptions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2830 (KP-3.32), 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 3.32. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.