[Paper Review] Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds

T. Sulanke, Frank H. Lutz|ArXiv.org|Sep 30, 2006

Computational Geometry and Mesh Generation46 references4 citations

TL;DR

This paper presents an isomorphism-free lexicographic enumeration algorithm for triangulated 2- and 3-manifolds, significantly improving efficiency over prior methods. The method generates all combinatorially distinct triangulations up to isomorphism by systematically exploring canonical forms, enabling the complete enumeration of all 11-vertex triangulated surfaces and 3-manifolds, including 17,263,8650 distinct 3-manifolds and 30 vertex-minimal triangulations of ℝℙ³.

ABSTRACT

We present a fast enumeration algorithm for combinatorial 2- and 3-manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3-manifolds with 11 vertices. We further determine all equivelar polyhedral maps on the non-orientable surface of genus 4 as well as all equivelar triangulations of the orientable surface of genus 3 and the non-orientable surfaces of genus 5 and 6.

Motivation & Objective

• To develop an efficient, isomorphism-free algorithm for enumerating combinatorial 2- and 3-manifolds up to isomorphism.
• To compute the complete list of triangulated surfaces with 11 and 12 vertices.
• To enumerate all triangulated 3-manifolds with 11 vertices and classify their topological and combinatorial types.
• To determine minimal vertex counts for triangulating specific 3-manifolds, including ℝℙ³ and lens spaces.
• To identify equivelar triangulations of specific surfaces, such as the non-orientable surface of genus 4 and the orientable surface of genus 3.

Proposed method

• The algorithm uses lexicographic enumeration in canonical form, ensuring each triangulation is generated only once up to isomorphism.
• It employs a depth-first search with pruning to avoid generating isomorphic copies during the enumeration process.
• The method systematically explores all possible sets of triangles (tetrahedra) in lexicographic order, maintaining combinatorial invariance.
• It leverages the fact that links of vertices must be PL spheres, using this to filter valid triangulations early in the search.
• The implementation, named lextri, uses optimized data structures and combinatorial checks to accelerate the enumeration of valid simplicial complexes.
• Topological types are determined heuristically for small triangulations, using Euler characteristic and orientability, with known algorithms for 3-sphere recognition.

Experimental results

Research questions

• RQ1 How many triangulated 2-manifolds exist with 11 and 12 vertices, up to combinatorial isomorphism?
• RQ2 How many triangulated 3-manifolds with 11 vertices exist, and what are their topological and combinatorial types?
• RQ3 What is the minimal number of vertices required to triangulate specific 3-manifolds such as ℝℙ³ and lens spaces?
• RQ4 Which equivelar triangulations exist for specific surfaces, including the non-orientable surface of genus 4 and the orientable surface of genus 3?
• RQ5 Are all 3-spheres with 11 vertices shellable, and what is the smallest known non-shellable 3-sphere?

Key findings

• There are exactly 17,263,8650 distinct triangulated 3-manifolds with 11 vertices.
• The 3-sphere S³ has 166,564,303 triangulations with 11 vertices, while ℝℙ³ has exactly 30 vertex-minimal triangulations with 11 vertices.
• The minimal triangulation of ℝℙ³ requires 11 vertices, and Walkup’s triangulation with f-vector (11,51,80,40) is the unique vertex- and facet-minimal one.
• All 3-spheres with 11 vertices are shellable, and the smallest known non-shellable 3-sphere has 13 vertices.
• There are 18,313,63502 triangulated 3-balls with 10 vertices, of which 277,479 are non-shellable.
• The minimal number of vertices for triangulating the connected sum (S²×S¹)#(S²×S¹) and (S²×̃S¹)#(S²×̃S¹) is 12.