On the Atiyah problem on hyperbolic configurations of four points

TL;DR

This paper proves Atiyah's conjecture on the linear independence of associated polynomials for certain configurations of four points in hyperbolic 3-space, confirming the conjecture in specific cases.

Contribution

It establishes the linear independence of Atiyah's polynomials for four-point configurations in hyperbolic space in two particular cases, advancing understanding of the conjecture.

Findings

Proves Atiyah's conjecture for non-coplanar four points.

Proves Atiyah's conjecture when one point lies in the convex hull of the other three.

Abstract

Given a configuration $\mathbf{x}$ of $n$ distinct points in hyperbolic $3$-space $H^3$, Michael Atiyah associated $n$ polynomials $p_1,\ldots,p_n$ of a variable $t \in \mathbb{C}P^1$, of degree $n-1$, and conjectured that they are linearly independent over $\mathbb{C}$, no matter which configuration $\mathbf{x}$ one starts with. We prove this conjecture for $n=4$ in two cases: in case the $4$ points are non-coplanar, and in case one of the points lies in the hyperbolic convex hull of the other three.