Kirby Problem 1.3

Definitions and mathematical statement

For a smooth knot $K\subset S^3$, the unknotting number $u(K)$ is the least number of crossing changes needed to obtain the unknot, minimized over all diagrams of $K$. A crossing change reverses which of the two strands passes over the other; ambient isotopies between changes are allowed. A knot is nontrivial if it is not ambiently isotopic to the round circle. Write $K_1\#\cdots\#K_n$ for connected sum.

A Conway mutation cuts a knot along a sphere meeting it transversely in four points, rotates one of the resulting two-string tangles through a half-turn preserving its endpoint set, and reglues. When the result is a knot, call it $K'$. The questions are:

1. For every $n\geq1$ and every collection of nontrivial knots $K_i$, is $u(K_1\#\cdots\#K_n)\geq n$?
2. For every pair of mutant knots, is $u(K')=u(K)$?

Part (a) is a lower bound by the number of summands. It makes no assertion that unknotting number is additive, and therefore retains the distinct target of this catalogue entry.

Short English statement

Must a sum of $n$ nontrivial knots need at least $n$ crossing changes, and does mutation preserve unknotting number?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2662 (KP-1.3), 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 1.3. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.