[Paper Review] Verification and Strengthening of the Atiyah--Sutcliffe Conjectures for Several Types of Configurations

Dragutin Svrtan, Igor Urbiha|ArXiv.org|Sep 6, 2006

Advanced Combinatorial Mathematics10 references3 citations

TL;DR

This paper verifies and strengthens the Atiyah–Sutcliffe conjectures C2 and C3 for various geometric configurations, including parallelograms, cyclic quadrilaterals, and specific tetrahedra, using multi-Schur functions and algebraic identities. It proves the conjectures up to n=9 and proposes new conjectures for almost collinear configurations, linking them to physical theories like Skyrmions and Fullerenes.

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.Đoković). 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 Đokovi\' 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.

Motivation & Objective

• To verify the Atiyah–Sutcliffe conjectures C2 and C3 for specific geometric configurations such as parallelograms, cyclic quadrilaterals, and symmetric tetrahedra.
• To propose and investigate strengthened versions of conjecture C3 for four-point configurations, including symmetric and almost collinear arrangements.
• To extend the verification of the conjectures up to n=9 using symbolic computation with multi-Schur functions.
• To explore connections between the geometric energy defined by the Atiyah determinant and physical theories such as Skyrmions and Fullerenes.
• To establish new algebraic inequalities involving the d3 function and Schur-type symmetric polynomials, supporting the conjectures.

Proposed method

• Construct the Atiyah determinant Dn from unit vectors between n points in R³, mapped to the Riemann sphere via stereographic projection.
• Define n polynomials pi of degree n−1 with roots at the projected directions tij, and form the matrix Mn with coefficients of these polynomials.
• Compute the determinant Dn = det(Mn), and normalize it to obtain the normalized Atiyah determinant D̃n.
• Use multi-Schur functions and symbolic computation (via Stembridge’s SF package) to express and factorize differences in symmetric function expansions.
• Apply Schur’s inequality and its generalizations to prove nonnegativity of coefficients in symmetric function expansions, supporting conjectures.
• Verify conjectures numerically and algebraically up to n=9 by substituting variable increments and checking nonnegative coefficients.

Experimental results

Research questions

• RQ1 Does the Atiyah–Sutcliffe conjecture C2 hold for all configurations of four points in the plane?
• RQ2 Can the stronger conjecture C3 be verified or strengthened for symmetric four-point configurations such as parallelograms and cyclic quadrilaterals?
• RQ3 What are the implications of the geometric energy E_n = -log|Dn| for configurations with dihedral symmetry or near-collinearity?
• RQ4 Do the proposed new conjectures for almost collinear configurations imply the original C2 and C3 conjectures?
• RQ5 Can the algebraic structure of the d3 function and its inequalities be used to prove the non-vanishing of the Atiyah determinant for general n?

Key findings

• The Atiyah–Sutcliffe conjectures C2 and C3 are verified for all parallelograms, cyclic quadrilaterals, and several infinite families of tetrahedra.
• The normalized Atiyah determinant D̃n is shown to be non-zero for all tested configurations up to n=9, supporting the conjecture that |Dn| ≥ 1.
• A strengthened version of conjecture C3 is proposed for four-point configurations, based on symmetric function identities and verified via multi-Schur function expansions.
• For almost collinear configurations, new conjectures involving symmetric functions are introduced, which imply C2 and C3.
• The computation using multi-Schur functions and variable substitutions (Xk = Xk+1 + hk) confirms that all coefficients in the expansion are nonnegative, supporting the conjectures algebraically.
• The paper establishes new symmetric function inequalities, including d3(x,y,z) ≤ 4x²y²z² / (xyz + x³ + y³ + z³), which support the conjectures and generalize Schur’s inequality.