The complete $10$-tetrahedra census of orientable cusped hyperbolic$3$-manifolds

This paper extends the complete census of orientable cusped hyperbolic 3-manifolds to 10 tetrahedra, identifying 150,730 new manifolds and their triangulations to determine 439,898 exceptional Dehn fillings, discover 1,849 new simplest hyperbolic knot exteriors, and provide the first example of such a manifold containing a closed totally geodesic surface.

The Gist

Imagine the universe of shapes as a giant, infinite library. For over a century, mathematicians have been trying to catalog every single book in a specific, very strange section of this library: hyperbolic 3-manifolds.

Think of these manifolds as "space bubbles." Unlike the flat space we live in (where parallel lines never meet), these bubbles are shaped like a saddle or a Pringles chip that curves everywhere. They are "cusped," meaning they have holes in them that stretch out infinitely, like the inside of a donut that goes on forever.

For a long time, this library was only partially organized. Mathematicians had cataloged all the "books" (manifolds) that could be built using up to 9 tetrahedra. A tetrahedron is just a pyramid with a triangular base (like a slice of a tetrahedral cheese).

Shana Yunsheng Li, a mathematician from the University of Illinois, has just finished a massive project to catalog the next shelf: all the shapes that can be built using exactly 10 tetrahedra.

Here is the breakdown of what this paper achieved, explained through simple analogies:

1. The Great Expansion (The Census)

Imagine you are building a city out of LEGO bricks.

• The Old Census: Previously, researchers had listed every possible city you could build with up to 9 bricks. There were about 44,000 of them.
• The New Census: Li built the list for cities made of 10 bricks.
• The Result: The number exploded! There are 150,730 unique cities (manifolds) in this new category.
• The Blueprints: Each city can be built in different ways. Li found 496,638 different sets of blueprints (triangulations) to build these 150,730 cities.

Why is this hard?
It's like trying to find every unique snowflake that can be made with 10 ice crystals. As you add just one more crystal, the number of possibilities grows exponentially. To do this, Li had to use supercomputers to generate millions of candidates and then filter them out.

2. The "Magic Mirror" (Verification)

How do you know two different-looking blueprints actually build the exact same city?
In the past, mathematicians used "algebraic mirrors" (math formulas) to check if two shapes were the same. But these mirrors were sometimes blurry; they might say two shapes were different when they were actually the same, or vice versa, because of tiny calculation errors.

Li introduced a "Verified Magic Mirror."

• Instead of just guessing the shape, this new method uses rigorous, error-free math (like a super-precise ruler) to calculate the "fingerprint" of the shape.
• This fingerprint is called the Canonical Triangulation. It's like a unique DNA test for these space bubbles. If the DNA matches, the shapes are identical. If it doesn't, they are different.
• This new method allowed Li to sort the millions of candidates into the correct 150,730 groups without any duplicates.

3. The Applications (What can we do with this list?)

Now that we have this massive catalog, we can use it to solve other puzzles:

• The "Pop" Test (Dehn Fillings): Imagine taking one of these infinite space bubbles and plugging the holes with a cork (a process called Dehn filling). Sometimes, plugging the hole makes the bubble collapse into a non-hyperbolic shape (like a sphere). Li found 439,898 specific ways to plug these holes that cause a "pop" (an exceptional filling). This helps us find the simplest knots in our own universe (the 3-sphere).
• The "Fiber" Check: Some of these shapes can be thought of as a bundle of fibers (like a deck of cards). Li checked if the shapes in the new list could be "unrolled" into a flat sheet. For almost all of them, the math held up perfectly, confirming a major theory about how these shapes behave.
• The "Hidden Surface" Hunt: Some of these space bubbles contain a hidden, flat, closed surface inside them (like a flat sheet of paper floating inside a balloon). For a long time, no one could find one in the smaller catalogs (9 tetrahedra or less).
  • The Discovery: Li found the simplest example of such a shape in the new 10-tetrahedra list. It's a shape called o10_143602. It's the "smallest" space bubble known to contain a hidden, perfectly flat, closed surface.

4. The Future (11 Tetrahedra)

The paper ends with a teaser. Li has already started working on the next shelf: 11 tetrahedra.

• Generating the list for 10 tetrahedra took about 2 months on a single computer thread (or 300 threads working together).
• The list for 11 tetrahedra is so huge that it would take a single computer thread 6 years to generate.
• Li has already generated the candidates for 11 tetrahedra and is preparing a new paper to release that data.

Summary

Shana Yunsheng Li has essentially updated the "Periodic Table" of 3D space shapes. By using a new, ultra-precise way to check if shapes are identical, they cataloged over 150,000 new complex shapes. This isn't just a list; it's a toolkit that allows mathematicians to test deep theories about knots, the structure of the universe, and the nature of space itself.

Technical Summary

Here is a detailed technical summary of the paper "The complete 10-tetrahedra census of orientable cusped hyperbolic 3-manifolds" by Shana Yunsheng Li.

1. Problem Statement

The paper addresses the enumeration of orientable cusped hyperbolic 3-manifolds. While the census of such manifolds up to 9 tetrahedra was completed by Burton in 2014 (yielding 44,250 manifolds), progress had stalled due to the exponential growth in computational complexity and the difficulty of distinguishing isometry classes (i.e., determining when two different triangulations represent the same manifold).

The primary goal is to extend this census to 10 tetrahedra, providing a complete list of all such manifolds, their minimal ideal triangulations, and analyzing their topological properties. A secondary goal is to resolve issues regarding the certification of hyperbolicity and the existence of geometric triangulations.

2. Methodology

The author employs a three-step pipeline standard in census construction: (i) Candidate Generation, (ii) Eligibility Testing, and (iii) Grouping and Separation.

A. Candidate Generation

• Tool: Used Regina's tricensus utility.
• Scope: Generated all ideal triangulations with 10 tetrahedra where every vertex is ideal, and no edges have degree 1, 2, or degree 3 (if the edge belongs to three distinct tetrahedra).
• Initial Count: 8,373,308 triangulations (including non-orientable ones).

B. Eligibility Testing (Filtering)

Candidates were filtered through four rigorous tests to ensure they represent minimal ideal triangulations of cusped hyperbolic 3-manifolds:

1. Greedy Non-minimality: Uses simplification algorithms to check if the triangulation can be reduced to fewer tetrahedra.
2. Exhaustive Non-minimality: Performs exhaustive searches of Pachner moves (2-3 and 3-2) within a bounded complexity to ensure no simpler triangulation exists.
3. Special Surfaces: Checks for normal surfaces that would violate minimality or hyperbolicity conditions (e.g., positive Euler characteristic, or specific one-sided surfaces).
4. Certify Hyperbolicity: Uses SnapPy to attempt finding a complete hyperbolic structure. If a structure is found, the manifold is hyperbolic.

After these steps, 904,452 triangulations were certified as representing cusped hyperbolic 3-manifolds.

C. Grouping and Separation (The Core Innovation)

The most significant methodological contribution is the use of Verified Canonical Triangulations to deduplicate the census.

• The Challenge: Previous methods relied on numerical computation of the Epstein–Penner canonical cellulation. Numerical errors (e.g., misidentifying "flat" tetrahedra or incorrect signs of "tilt" parameters) could lead to incorrect canonical forms, causing distinct manifolds to be grouped together or vice versa.
• The Solution: The author utilized Verified Computation (implemented by M. Goerner in SnapPy).
  • Interval Arithmetic: Used to rigorously compare quantities (e.g., determining if a tilt is strictly positive or negative) to avoid floating-point errors.
  • LLL Algorithm: Used to symbolically solve trace fields, representing them as number fields to prove equality of algebraic invariants.
• Process:
  1. Computed verified canonical triangulations for the 608,918 orientable candidates.
  2. 16 candidates failed verification due to complex trace fields; these were separated using high-precision volume calculations and homology groups.
  3. The verified canonical triangulation acts as a complete invariant for these manifolds, allowing for exact deduplication.

3. Key Contributions and Results

A. The Main Census (Theorem 1)

• Manifolds: There are exactly 150,730 distinct orientable cusped hyperbolic 3-manifolds with minimal ideal triangulations of 10 tetrahedra.
• Triangulations: These manifolds possess a total of 496,638 minimal ideal triangulations.
• Data Availability: The full dataset is integrated into SnapPy 3.3 and available via DOI.

B. Geometric Triangulations (Corollary 2 & Proposition 11)

• Result: All cusped hyperbolic 3-manifolds in this census (and those with $\le 9$ tetrahedra) admit geometric triangulations.
• Significance: This resolves a theoretical uncertainty about whether all such manifolds admit geometric triangulations within this complexity bound, as the verified computation confirms the existence of a geometric structure for every entry.

C. Exceptional Dehn Fillings (Theorem 14)

• The paper identifies 439,898 exceptional Dehn fillings (fillings that result in non-hyperbolic manifolds) on the 1-cusped manifolds in the census.
• This reveals 1,849 new simplest hyperbolic knot exteriors in $S^3$ (Theorem 15).

D. Thurston Norm and Twisted Alexander Polynomials (Section 4.2)

• For 143,898 manifolds with $b_1(M)=1$, the author verified that the inequality $x(M) \ge \frac{1}{2} \deg \tau_{2, \tilde{\rho}}(M)$ is an equality for all cases where the manifold fibers over $S^1$.
• Confirmed that 9 specific manifolds do not fiber over $S^1$ because their twisted Alexander polynomials are not monic.

E. L-Space Conjecture (Section 4.3)

• Generated a list of 1,899,974 pairs $(M, \alpha)$ where $M(\alpha)$ is a hyperbolic $\mathbb{Q}$-homology 3-sphere with systole length $\ge 0.163$. This provides a massive dataset for testing the L-space conjecture (equivalence of left-orderability, non-minimal Heegaard Floer homology, and taut foliations).

F. Totally Geodesic Surfaces (Theorem 18)

• Discovered the simplest orientable cusped hyperbolic 3-manifold containing a closed totally geodesic surface: $o10\_143602$.
• Properties: 2-cusped, volume $\approx 9.1345$, $H_1 = \mathbb{Z}_2 \oplus \mathbb{Z}_6$.
• Surface: Contains exactly one such surface (up to isotopy), which is non-orientable with Euler characteristic $-1$.
• Significance: No such manifold existed in the 9-tetrahedra census.

4. Significance and Future Outlook

• Computational Topology: This work represents a major leap in low-dimensional topology, extending the census by a factor of roughly 3.4 (from 44k to 150k manifolds).
• Methodological Shift: The successful application of verified computation (combining interval arithmetic and symbolic algebra) overcomes the limitations of purely numerical methods. This technique is presented as a viable path for extending the census to 11 tetrahedra and beyond.
• Scalability: The author notes that generating 11-tetrahedra candidates is feasible (taking ~6 years on a single thread) and that the 11-tetrahedra orientable census has already been completed (Theorem 12), with 27,794,289 candidates identified.
• Obstacles: The primary bottleneck remains the exponential growth of candidate generation (Step i). While the separation step (Step iii) is now robust via verified computation, the generation step requires massive parallel computing resources.

In summary, this paper provides the definitive census for 10-tetrahedra manifolds, introduces a rigorous verification framework that eliminates numerical ambiguity, and unlocks new avenues for research in knot theory, Dehn surgery, and geometric structures.