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.
Categories: math.MG math.GT