# Seidel's conjectures in hyperbolic 3-space

## TL;DR

This paper proves two conjectures by Seidel regarding the volume of ideal hyperbolic tetrahedra, showing it depends monotonically on algebraic properties of the Gram matrix of vertices in hyperbolic 3-space.

## Contribution

It establishes the relationship between the volume of ideal hyperbolic tetrahedra and algebraic invariants of their Gram matrices, confirming Seidel's conjectures.

## Key findings

- Volume determined by Gram matrix invariants
- Volume is a monotonic function of matrix determinants and permanents
- Proof of Seidel's conjectures in hyperbolic 3-space

## Abstract

We prove, in the case of hyperbolic 3-space, a couple of conjectures raised by J. J. Seidel in "On the volume of a hyperbolic simplex", Stud. Sci. Math. Hung. 21, 243-249, 1986. These conjectures concern expressing the volume of an ideal hyperbolic tetrahedron as a monotonic function of algebraic maps. More precisely, Seidel's first conjecture states that the volume of an ideal tetrahedron in hyperbolic 3-space is determined by (the permanent and the determinant of) the doubly stochastic Gram matrix $G$ of its vertices; Seidel's fourth conjecture claims that the mentioned volume is a monotonic function of both the permanent and the determinant of $G$.