Classical 3-manifold theory — Question 9.4

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 surface lamination $\Lambda$ is a closed subset of $M$ locally homeomorphic to a product of a disk with a closed transverse set. It is essential when it has no sphere leaf or torus leaf bounding a solid torus, and its complementary regions are irreducible, have incompressible boundary, and have no compressing monogon. Such a monogon is a compressing disk with one boundary arc on a boundary leaf and the other along a transverse interval fiber. It is genuine if some complementary region is not an interval bundle. A sublamination is a closed union of leaves. An alternating knot has a projection with alternating over- and undercrossings along it; a torus knot lies on an unknotted torus. A polygon-bundle complementary region is a bundle over $S^1$ whose fiber is an ideal polygon. A monkey-saddle filling uses saddle leaves asymptotic to consecutive boundary leaves in alternating transverse directions; an even number of sides permits this alternation.

Statement recorded in the supplied source.

For each non-torus alternating knot $K$, can essential laminations realizing the nontrivial boundary slopes be arranged with an even-sided polygon-bundle complementary region containing $K$, so that every nontrivial surgery on $K$ admits the monkey-saddle filling? The requirement is the extra complementary-region structure and its filling, beyond merely realizing boundary slopes. [S2]

Short English statement

Strengthen the lamination constructions for non-torus alternating knots so every nontrivial surgery can be filled using an even-sided region and monkey saddles.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300037 (AMR-102-0037). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Danny Calegari, Problems in foliations and laminations of 3-manifolds, version 0.78 (2002), Question 9.4; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.