Existence questions — Question 2.2

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 codimension-one foliation $\mathcal F$ decomposes a $3$-manifold $M$ into surface leaves, locally as the horizontal slices of a product chart. In the terminology of the source, it is taut if a closed curve transverse to $\mathcal F$ meets every leaf. A minimal foliation means that every leaf is dense; a minimal surface means a surface with vanishing mean curvature for the specified Riemannian metric. These two uses of minimal are distinct. For a finite-depth foliation, depth-zero leaves are compact; a leaf has depth $d>0$ when its frontier is a union of leaves of smaller depth and $d$ is the smallest such finite bound. The depth of the foliation is the greatest leaf depth. For a knot, depth is the minimum depth of a taut foliation of its exterior with the standard boundary behavior used for knot-depth foliations; fibered knots have depth zero.

Statement recorded in the supplied source.

For every prescribed nonnegative integer $n$, find effective algorithms to construct a hyperbolic knot of depth exactly $n$ and to recognize that its depth is $n$. Ask also for construction and recognition with the requirement replaced by depth at least $n$. Hyperbolic means that the knot complement admits a complete finite-volume hyperbolic metric. [S2]

Short English statement

Construct and recognize hyperbolic knots of any prescribed foliation depth, and also of depth at least a prescribed integer.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300002 (AMR-102-0002). 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 2.2; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.