Branched surfaces and triangulations — Question 7.3

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 branched surface $B\subset M$ is a smoothly tangent $2$-complex with the standard local branching models. Its fibered neighborhood $N(B)$ is an interval bundle with the usual identifications over the branch locus. A lamination is carried by $B$ if it lies in $N(B)$ transverse to interval fibers, and fully carried if it meets every fiber. Splitting means cutting this fibered neighborhood along carried surface pieces and taking the induced branched surface. 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. A taut ideal triangulation consists of ideal tetrahedra with taut face coorientations (two incoming and two outgoing faces per tetrahedron, with the taut compatibility around edges). Its smoothed faces determine a carrying branched surface. A slope on a torus boundary is an unoriented direction, with rational slopes represented by primitive classes in $H_1(T^2;\mathbb Z)$ up to sign; slope terminology extends to carried boundary laminations.

Statement recorded in the supplied source.

For a fixed taut ideal triangulation, characterize all boundary slopes realized by essential laminations carried by its associated branched surface, and provide an effective algorithm to determine them. Include the full class of carried essential laminations, not only a preferred lamination singled out by extra triangulation structure. [S2]

Short English statement

Determine every boundary slope carried by essential laminations of a given taut ideal triangulation, and give an algorithm.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300018 (AMR-102-0018). 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 7.3; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.