[Paper Review] Combinatorial decompositions, Kirillov-Reshetikhin invariants and the Volume Conjecture for hyperbolic polyhedra

Alexander Kolpakov, Jun Murakami|arXiv (Cornell University)|Mar 7, 2016

Geometric and Algebraic Topology14 references4 citations

TL;DR

This paper proposes a combinatorial method to compute the hyperbolic volume of simple 3D polyhedra by decomposing them into generalized tetrahedra via I-H and capping moves, linking the volume to critical values of a potential function. It further conjectures a deep connection between this geometric decomposition and the asymptotic behavior of Kirillov-Reshetikhin invariants of the polyhedron’s 1-skeleton, offering a potential pathway to proving the Volume Conjecture for hyperbolic polyhedra.

ABSTRACT

We suggest a method of computing volume for a simple polytope $P$ in three-dimensional hyperbolic space $\\mathbb{H}^3$. This method combines the combinatorial reduction of $P$ as a trivalent graph $\\Gamma$ (the $1$-skeleton of $P$) by $I-H$, or Whitehead, moves (together with shrinking of triangular faces) aligned with its geometric splitting into generalised tetrahedra. With each decomposition (under some conditions) we associate a potential function $\\Phi$ such that the volume of $P$ can be expressed through a critical values of $\\Phi$. The results of our numeric experiments with this method suggest that one may associated the above mentioned sequence of combinatorial moves with the sequence of moves required for computing the Kirillov-Reshetikhin invariants of the trivalent graph $\\Gamma$. Then the corresponding geometric decomposition of $P$ might be used in order to establish a link between the volume of $P$ and the asymptotic behaviour of the Kirillov-Reshetikhin invariants of $\\Gamma$, which is colloquially know as the Volume Conjecture.

Motivation & Objective

• To develop a systematic method for computing the hyperbolic volume of simple 3D polyhedra in $\mathbb{H}^3$ using combinatorial decompositions.
• To establish a geometric link between the volume of a hyperbolic polyhedron and the asymptotic behavior of Kirillov-Reshetikhin invariants of its 1-skeleton as a trivalent graph.
• To explore whether the sequence of I-H and capping moves used in volume computation corresponds to the sequence used in computing Kirillov-Reshetikhin invariants.
• To provide numerical evidence supporting the Volume Conjecture for hyperbolic polyhedra through explicit computations on prisms and dodecahedra.

Proposed method

• Decompose a simple hyperbolic polyhedron $P$ into generalized tetrahedra $T_i$ such that $\mathrm{Vol}(P) = \sum \mathrm{Vol}(T_i)$.
• Apply a sequence of I-H (Whitehead) moves and capping moves to reduce the 1-skeleton $\Gamma$ of $P$ to a tetrahedron, preserving geometric and combinatorial structure.
• Associate each decomposition with a potential function $\Phi(\ell_1, \dots, \ell_m)$ depending on lengths of common perpendiculars between faces, where the volume is determined by critical values $\Phi(\ell^*_1, \dots, \ell^*_m)$.
• Use numerical implementations in Wolfram Mathematica® and cross-validate with Orb software to compute volumes and Kirillov-Reshetikhin invariants.
• Define edge colorings of $\Gamma$ based on dihedral angles of $P$ to compute Kirillov-Reshetikhin invariants $\langle \Gamma, c^{(r)} \rangle$ for odd $r$.
• Conjecture that the asymptotic behavior of these invariants as $r \to \infty$ matches the volume of $P$, via limits of quantum $6j$-symbols.

Experimental results

Research questions

• RQ1 Can the hyperbolic volume of a simple polyhedron in $\mathbb{H}^3$ be computed via a sequence of I-H and capping moves applied to its 1-skeleton?
• RQ2 Is there a correspondence between the sequence of moves used to reduce the 1-skeleton of a polyhedron and the sequence used to compute its Kirillov-Reshetikhin invariants?
• RQ3 Does the asymptotic behavior of Kirillov-Reshetikhin invariants of the 1-skeleton $\Gamma$ converge to the hyperbolic volume of the original polyhedron $P$?
• RQ4 Can the volume of a generalized hyperbolic tetrahedron be recovered from the limit of quantum $6j$-symbols associated with its dihedral angles?
• RQ5 Is the Volume Conjecture for hyperbolic polyhedra supported by numerical evidence from explicit computations on prisms and dodecahedra?

Key findings

• The volume of a hyperbolic polyhedron $P$ can be expressed as the critical value of a potential function $\Phi$ derived from geometric parameters of a decomposition into generalized tetrahedra.
• For a regular ideal tetrahedron with dihedral angle $\alpha$, the limit $2\pi \lim_{r\to\infty} \frac{1}{r} \log \left| \left\{ \begin{smallmatrix} k & k & k \\ k & k & k \end{smallmatrix} \right\}_{q=\exp(4\pi i / r)}^{RW} \right| = \mathrm{Vol}(T_\alpha)$, matching the volume of the tetrahedron.
• For a doubly truncated tetrahedron with dihedral angles $\alpha, \beta$, the volume is recovered as $2\pi \lim_{r\to\infty} \sum_{j\,\text{odd}} \frac{1}{r} \log \left| \left\{ \begin{smallmatrix} a^{(r)} & b^{(r)} & e^{(r)} \\ d^{(r)} & c^{(r)} & \frac{j-1}{2} \end{smallmatrix} \right\}_{q=\exp(4\pi i / r)}^{RW} \cdot \frac{\{(j+1)(2f^{(r)}+1)\}}{\{j\}} \right| = \mathrm{Vol}(T)$.
• Numerical experiments on a pentagonal prism $\Pi$ and a pleated prism $\mathscr{P}$ confirm consistency between volume computations via decomposition and known software (Orb).
• The method successfully computes volumes for a dodecahedron with Coxeter dihedral angles, demonstrating applicability beyond Coxeter polytopes.
• The authors observe strong numerical agreement between the asymptotic behavior of Kirillov-Reshetikhin invariants and the hyperbolic volume, supporting the Volume Conjecture for hyperbolic polyhedra.