Filling a triangulation of the 2-sphere

TL;DR

The paper resolves the tetvol conjecture for all $v \,>=\, 13$ by a purely combinatorial, LP/flow-inspired approach that replaces STT's hyperbolic-volume method with volume potentials on carefully constructed phyllohedra derived from Eisenstein lattices. It introduces the $Qvol$ framework and the $(5,1)$-phyllohedra family $T_v$, showing existence of good volume potentials $\rho_v$ with $\rho_v(T_v) = 2v-10$ (verified computationally), thereby proving $tetvol(T_v) = 2v-10$ and yielding implications for the maximum rotation/flip distance $d'(v)$. The work discusses the broader LP viewpoint, explores additional phyllohedra families and their volume bounds, and clarifies the relation between tetvol and flip distance, including limitations illustrated by known gaps. It also confirms that a minimal tetration of a spherical triangulation fills a 3-ball and situates the results within the larger program initiated by MT, pointing to further generalizations via phyllohedra and LP-based methods.

Abstract

Define the tet-volume of a triangulation of the 2-sphere to be the minimum number of tetrahedra in a 3-complex of which it is the boundary, and let $d(v)$ be the maximum tet-volume for $v$-vertex triangulations. In 1986 Sleator, Tarjan, and Thurston (STT) proved that $d(v) = 2v-10$ holds for large $v$, and conjectured that it holds for all $v \geq 13$. Their proof used hyperbolic polyhedra of large volume. They suggested using more general notions of volume instead. In work that was all but lost, Mathieu and Thurston used this approach to outline a combinatorial proof of the STT asymptotic result. Here we use a much simplified version of their approach to prove the full conjecture. This implies STT's weaker conjecture, proven by Pournin in 2014, characterizing the maximum rotation distance between trees.

Figures (14)

• Figure 2: Code to define $T_v$ and check that $\rho(T_v)=2v-10$.
• Figure 3: Triation $T_{23}$
• Figure 4: Triations $T_{13}$ through $T_{22}$
• Figure 5: $T_{13}$ and $T_{14}$ unwrapped
• Figure 6: The phyllocylinders $\Phi_{6,0},\Phi_{5,1},\Phi_{4,2},\Phi_{4,2},\Phi_{5,2}$. Unless $a$ or $b$ vanishes, $\Phi_{a,b}$ has $b$ spirals in direction $1$ (black); $a$ spirals in direction $\omega$ (red); $a+b$ spirals in direction $1+\omega$ (blue).

Theorems & Definitions (1)

• Conjecture : tetvol