Generalizations of the Conway-Gordon theorems and intrinsic knotting on complete graphs

TL;DR

This paper extends Conway-Gordon theorems to complete graphs with any number of vertices ≥6 and explores their implications for intrinsic knotting and linking in spatial graphs.

Contribution

It generalizes integral Conway-Gordon theorems to arbitrary-sized complete graphs and relates knot invariants to linking configurations.

Findings

Generalized integral Conway-Gordon theorems for complete graphs with ≥6 vertices.

Explicit formulas for the sum of Conway polynomial coefficients in rectilinear spatial graphs.

Connections between knot invariants and linking numbers in complex spatial graphs.

Abstract

In 1983, Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and that for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is odd. In 2009, the second author gave integral lifts of the Conway-Gordon theorems in terms of the square of the linking number and the second coefficient of the Conway polynomial. In this paper, we generalize the integral Conway-Gordon theorems to complete graphs with arbitrary number of vertices greater than or equal to six. As an application, we show that for every rectilinear spatial complete graph whose number of vertices is greater than or equal to six, the sum of the second coefficients of the Conway polynomials over all of the Hamiltonian knots is determined explicitly in terms of the number of triangle-triangle Hopf links.