Kirby Problem 1.5

Definitions and mathematical statement

A Seifert surface for an oriented knot $K\subset S^3$ is a compact, connected, oriented embedded surface whose oriented boundary is $K$. Its least possible genus is the Seifert genus $g(K)$. The unknotting number $u(K)$ minimizes the number of crossing changes turning $K$ into the unknot, over all diagrams. A knot is fibered if its exterior $S^3\setminus\operatorname{int}\nu K$ is a bundle over $S^1$ with fiber a Seifert surface. It is positive if it admits an oriented diagram in which every crossing has positive sign under the usual right-hand convention. It is alternating if it admits a diagram whose overpasses and underpasses alternate along the knot.

Two questions impose different diagrammatic hypotheses:

1. Does every positive fibered knot satisfy $g(K)=u(K)$?
2. Does every alternating fibered knot satisfy $g(K)\geq u(K)$?

In each part the minimum defining $u(K)$ is unrestricted: the diagrams used in an unknotting sequence need not be positive, alternating, or diagrams of a fibered knot. Genus here means Seifert genus in $S^3$.

Short English statement

Compare genus and unknotting number for positive fibered knots and for alternating fibered knots.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2664 (KP-1.5), 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 1.5. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.