Berge Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A knot $K\subset S^3$ is a smooth embedded circle, considered up to ambient isotopy. Let $N(K)$ be a closed tubular neighborhood, a solid torus, and $X=S^3\setminus\operatorname{int}N(K)$ its exterior. A slope on the boundary torus is an unoriented isotopy class of essential simple closed curves. Dehn surgery along a slope $r$ glues a solid torus to $X$ so its meridian becomes $r$; write $S^3_r(K)$ for the result. The meridional slope, which restores $S^3$, is excluded.

A lens space $L(p,q)$ has $p\ge2$, $\gcd(p,q)=1$, and is the quotient of $S^3\subset\mathbb C^2$ by the free cyclic action generated by $(z_1,z_2)\mapsto(\zeta z_1,\zeta^q z_2)$, where $\zeta=e^{2\pi i/p}$. Thus $S^3$ and $S^1\times S^2$ are not lens spaces in this statement.

A knot is doubly primitive if it can be isotoped onto a genus-two Heegaard surface $\Sigma$ in $S^3=H_1\cup_\Sigma H_2$ such that, for each genus-two handlebody $H_i$, there is a properly embedded compressing disk whose boundary meets the knot transversely once. A genus-two handlebody is a ball with two one-handles, and a compressing disk has boundary essential in its boundary surface. Berge's enumerated knot families are precisely the doubly primitive knots, as established in Source S2; this supplies a geometric definition of the Berge-knot class.

Statement recorded in the supplied source.

For every knot $K\subset S^3$, if some nonmeridional Dehn surgery $S^3_r(K)$ is homeomorphic to a lens space, must $K$ be a Berge knot (equivalently, a doubly primitive knot)? The target classifies the original knot up to isotopy. Finding a Berge knot with the same surgery outcome or the same knot Floer homology does not establish that the original knot is Berge. [S2]

Short English statement

Are all knots in the three-sphere with a nontrivial lens-space surgery among Berge’s knot families?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1480 (TOP-002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Joshua Evan Greene, The Lens Space Realization Problem, arXiv:1010.6257v1 (2010), Section 1, Conjecture 1.1, Theorem 1.3 and Section 1.2 on Berge’s enumerated knots. https://arxiv.org/pdf/1010.6257v1.
• S3 Kenneth L. Baker, Dorothy Buck and Ana G. Lecuona, Some Knots in S1×S2 with Lens Space Surgeries, arXiv:1302.7011v1 (2013), Section 1, first two paragraphs on the S3 Berge conjecture. https://arxiv.org/pdf/1302.7011v1.