Kirby Problem 4.35

Definitions and mathematical statement

A two-knot is an embedding $k:S^2\hookrightarrow S^4$, either smooth or locally flat topological; address the two categories separately. Its exterior is $E_k=S^4\setminus\operatorname{int}\nu k(S^2)$, where $\nu k(S^2)\cong S^2\times D^2$. The exterior and the open complement have the same homotopy type. Its knot group is $G_k=\pi_1(E_k)$.

(a) Give intrinsic algebraic necessary and sufficient conditions on a group $G$ for $G\cong G_k$ for a two-knot in the standard $S^4$. This asks for a structural characterization, beyond restating existence of a knot diagram or a specially realized presentation.

An orientation of the knot and ambient sphere identifies $H_1(E_k;\mathbb Z)\cong\mathbb Z$, generated by a positive meridian. Let $\widetilde E_k$ be the infinite cyclic cover corresponding to the kernel of this abelianization. A generator $t$ of its deck transformations makes \[ \mathcal A_k=H_1(\widetilde E_k;\mathbb Z) \] a module over $\Lambda=\mathbb Z[t,t^{-1}]$.

(b) Classify the $\Lambda$-modules isomorphic to $\mathcal A_k$ for such knots, again distinguishing smooth and locally flat realizability. Reversing the meridian convention replaces the $t$ action by its inverse; specify whether this identification is allowed in a classification.

Integral module structure is required, rather than only its rationalization or its order polynomial. Likewise, realization in a homotopy four-sphere alone does not immediately give smooth realization in the standard sphere.

Short English statement

Characterize two-knot groups and integral Alexander modules in the standard four-sphere.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2911 (KP-4.35), 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 4.35. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.