The Four Color Theorem meets Shapes of Polyhedra
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Soggetti: | |
| Accesso online: | |
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| _version_ | 1866908961529135104 |
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| author | Schwartz, Richard Evan |
| author_facet | Schwartz, Richard Evan |
| contents | We consider solutions to the
$4$-color problem for the vertices of
sphere triangulations with
degree sequence $6,...,6,4,4,4,4,4,4$.
We sort these solutions into combinatorial
types and show that each generic type $τ$ is
parametrized by the set of integer lattice points inside a
$4$-dimensional rational polyhedral convex cone ${\cal C\/}_τ$.
There is an integral quadratic form $Q_τ$
on ${\cal C\/}_τ$
whose diagonal part, evaluated on a lattice point,
is $3$ times the number of triangles in the corresponding
triangulation. We relate this structure to
the octahedral stratum of Thurston's
moduli space of flat cone structures on the
sphere. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12059 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Four Color Theorem meets Shapes of Polyhedra Schwartz, Richard Evan Metric Geometry We consider solutions to the $4$-color problem for the vertices of sphere triangulations with degree sequence $6,...,6,4,4,4,4,4,4$. We sort these solutions into combinatorial types and show that each generic type $τ$ is parametrized by the set of integer lattice points inside a $4$-dimensional rational polyhedral convex cone ${\cal C\/}_τ$. There is an integral quadratic form $Q_τ$ on ${\cal C\/}_τ$ whose diagonal part, evaluated on a lattice point, is $3$ times the number of triangles in the corresponding triangulation. We relate this structure to the octahedral stratum of Thurston's moduli space of flat cone structures on the sphere. |
| title | The Four Color Theorem meets Shapes of Polyhedra |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2604.12059 |