Verification and Strengthening of the Atiyah--Sutcliffe Conjectures for Several Types of Configurations

TL;DR

This paper investigates, verifies, and strengthens the Atiyah--Sutcliffe conjectures for various point configurations in three-dimensional space, using computational methods and proposing new related conjectures.

Contribution

It extends verification of the conjectures to new configurations, proposes a strengthened conjecture, and introduces computational techniques for verification up to nine points.

Findings

Verified conjectures for parallelograms, cyclic quadrilaterals, and certain tetrahedra.

Proposed new conjectures for four-point configurations and almost collinear points.

Used multi-Schur functions to verify conjectures up to nine points.

Abstract

In 2001 Sir M. F. Atiyah formulated a conjecture C1 and later with P. Sutcliffe two stronger conjectures C2 and C3. These conjectures, inspired by physics (spin-statistics theorem of quantum mechanics), are geometrically defined for any configuration of points in the Euclidean three space. The conjecture C1 is proved for $n = 3, 4$ and for general $n$ only for some special configurations (M. F. Atiyah, M. Eastwood and P. Norbury, D.{\DJ}okovi\'{c}). Interestingly the conjecture C2 (and also stronger C3) is not yet proven even for arbitrary four points in a plane. So far we have verified the conjectures C2 and C3 for parallelograms, cyclic quadrilaterals and some infinite families of tetrahedra. We have also proposed a strengthening of conjecture C3 for configurations of four points (Four Points Conjectures). For almost collinear configurations (with all but one point on a line) we propose several new conjectures (some for symmetric functions) which imply C2 and C3. By using computations with multi-Schur functions we can do verifications up to $n=9$ of our conjectures. We can also verify stronger conjecture of {\DJ}okovi\' c which imply C2 for his nonplanar configurations with dihedral symmetry. Finally we mention that by minimizing a geometrically defined energy, figuring in these conjectures, one gets a connection to some complicated physical theories, such as Skyrmions and Fullerenes.