A refinement of the Conway-Gordon theorems

TL;DR

This paper refines the Conway-Gordon theorems by providing integral versions involving linking numbers and Conway polynomial coefficients, offering new proofs for related theorems in spatial graph theory.

Contribution

It introduces integral lifts of the Conway-Gordon theorems and applies them to give alternative proofs of existing results for rectilinear spatial complete graphs.

Findings

Integral lifts of linking number and Conway polynomial coefficients.

Alternative topological proofs for Brown-Ramirez Alfonsin and Huh-Jeon theorems.

Refinement of classical spatial graph invariants.

Abstract

In 1983, Conway-Gordon showed that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this article, we give integral lifts of the Conway-Gordon theorems above in terms of the square of the linking number and the second coefficient of the Conway polynomial. As applications, we give alternative topological proofs of theorems of Brown-Ramirez Alfonsin and Huh-Jeon for rectilinear spatial complete graphs which were proved by computational and combinatorial methods.