Kirby Problem 1.13

Definitions and mathematical statement

Let $Y$ be a closed oriented $3$-manifold and $K\subset Y$ a null-homotopic knot. Let $X=Y\setminus\operatorname{int}\nu K$ and let $F\subset X$ be an oriented Seifert surface. Zero surgery $Y_0(K)$ fills $X$ along the longitude $\partial F$; capping $F$ with a meridian disk of the filling gives a closed surface $\widehat F$. For a surface $A=\bigsqcup A_i$ put $\chi_-(A)=\sum_i\max(0,-\chi(A_i))$. The Thurston norm of a relative homology class is the minimum of $\chi_-$ over embedded oriented representatives. As in the source, a connected surface is taut if it minimizes this norm in a nonzero class, or if it is a sphere.

The two assertions to establish are:

1. If $Y_0(K)$ fibers over $S^1$ with fiber class $[\widehat F]$, then $X$ fibers over $S^1$ with fiber class $[F]\in H_2(X,\partial X;\mathbb Z)$.
2. If $F$ is taut in $X$, then $\widehat F$ is taut in $Y_0(K)$.

The prescribed fiber classes in the first assertion are part of the hypothesis and conclusion; the existence of an unrelated fibration does not suffice.

Short English statement

Does zero surgery preserve the tautness of a Seifert surface and detect a knot fibration in its specified homology class?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2672 (KP-1.13), 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 1.13. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.