Kirby Problem 1.80

Definitions and mathematical statement

A crossing change in a knot diagram interchanges its overpassing and underpassing strands. The unknotting number is \[ u(K)=\min_{D\text{ a diagram of }K} \min\{k:D\text{ is unknotted by }k\text{ crossing changes}\}. \] The minimum ranges over all diagrams, with unrestricted Reidemeister moves or ambient isotopies allowed between changes. It is a finite nonnegative integer, and equals zero exactly for the unknot.

Is $u$ computable? Precisely, does a single algorithm accept any finite combinatorial knot diagram, halt, and output this minimum integer? The source also singles out the weaker decision question: is there an algorithm that halts on every knot diagram and decides whether its knot has $u(K)=1$?

Searching diagrams and crossing changes can produce explicit upper bounds and certificates that $u(K)\leq k$. Such a search alone does not certify that no shorter sequence exists. The requested computation must establish the minimum and terminate on negative instances of the decision question as well. No bound on its running time is imposed.

Short English statement

Can unknotting number always be computed, or at least can one always decide whether it equals one?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2739 (KP-1.80), 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 1.80. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.