Kirby Problem 1.16

Definitions and mathematical statement

For a nonempty link $L\subset S^3$, its double branched cover $\Sigma_2(L)$ is the closed $3$-manifold with a map to $S^3$ that is two-to-one off $L$ and has local model $(z,t)\mapsto(z^2,t)$ along the preimage of $L$. Equivalently, take the connected double cover of the exterior determined by sending every meridian to $1\in\mathbb Z/2$, and complete it over the link. The covering manifold is considered without a specified deck involution.

The questions ask for the fibers of this construction:

1. Given a knot $K$, determine all knot types $K'$ with $\Sigma_2(K')\cong\Sigma_2(K)$.
2. Give explicit moves on links such that two links are connected by finitely many of the moves, together with isotopies, exactly when their double branched covers are homeomorphic.

Use homeomorphism without an orientation restriction in the displayed formulation; for an oriented version require orientation-preserving homeomorphisms throughout. A complete move description needs both soundness, meaning each move preserves the covering manifold, and completeness, meaning it connects every pair with the same cover.

Short English statement

Classify knots with the same double branched cover, and find a complete set of cover-preserving moves on links.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2675 (KP-1.16), 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 1.16. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.