Virtual-knot problem 32 — Generalize virtual knot theory to virtual 2-spheres in 4-space.

Definitions and mathematical statement

An ordinary two-knot in four-space is an embedding $f:S^2\hookrightarrow{\mathbb{R}}^4$, considered up to ambient isotopy in a specified category. The source asks to generalize virtual knot theory to virtual two-spheres in four-space. A precise proposed theory must specify its generalized objects and their equivalence, how ordinary embedded two-spheres are represented, and how it extends the virtual-crossing/stabilization idea from one-dimensional knots.

For a diagrammatic approach, one would need to identify the admissible projected surface data and any virtual local singularity labels, give the complete allowed local moves, and state how ordinary surface-diagram moves are incorporated. For a geometric approach, one would need to specify the ambient objects and stabilization/equivalence operations. These are alternative ways to make the construction concrete, not a choice imposed by the source. A resulting comparison with ordinary two-knot isotopy must be stated; the excerpt does not assert injectivity of such a comparison or prescribe a particular higher-dimensional diagram category.

The requested source surface is $S^2$, not an arbitrary-genus surface. Smooth, locally flat piecewise-linear or topological embedding conventions, orientation and the exact meaning of "generalize" are unstated. Consequently this is a programme to define a theory with justified equivalence rules, not an existence theorem about an already specified set of virtual two-spheres.

Short English statement

Develop a virtual analogue of embedded two-spheres in four-space, including precise objects, moves or stabilizations, and their relation to ordinary two-knot isotopy. The source does not prescribe the higher-dimensional virtual model.

Sources

• Local UnsolvedMath registry, permanent ID $10600032$, exact source code catalogue entry.
• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, problem-list item 32: https://arxiv.org/abs/1409.2823. Registry access 2026-07-29; source-tex extraction.
• The research summary mentions spinning constructions and Gauss-diagram invariants in scattered virtual-surface work, but supplies no specific references.