Kirby Problem 1.17

Definitions and mathematical statement

A link in $S^3$ is a finite disjoint union of smoothly embedded circles, considered up to ambient isotopy. It is alternating if it has a regular projection with crossing information in which overcrossings and undercrossings alternate as each component is traversed. A nonalternating link is one admitting no alternating diagram; possessing a single nonalternating diagram is not enough.

For a link $L$, let $\Sigma_2(L)$ be its double branched cover: away from $L$ the cover corresponds to the homomorphism of the link group to $\mathbb Z/2$ sending every meridian to the nonzero element, and along $L$ it is completed by the local branched model $(z,t)\mapsto(z^2,t)$.

Do there exist an alternating link $A$ and a nonalternating link $N$ with \[ \Sigma_2(A)\cong\Sigma_2(N)? \] The homeomorphism is between the underlying $3$-manifolds and need not intertwine the two covering involutions. Equivalently, does the existence of an alternating branch-set presentation force every branch-set presentation over $S^3$ of that same manifold to be alternating? Allowing orientation reversal does not affect the question, since mirroring preserves alternation.

Short English statement

Can one closed $3$-manifold be the double branched cover of both an alternating and a nonalternating link?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2676 (KP-1.17), 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 1.17. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.