On Conway-Gordon type theorems for graphs in the Petersen family

TL;DR

This paper extends known modulo 2 linking number results for Petersen family graphs to an integral formula involving squared linking numbers and Conway polynomial coefficients, deepening understanding of spatial graph invariants.

Contribution

It provides an integral lift of the Conway-Gordon type theorem for Petersen family graphs, connecting linking numbers with polynomial invariants.

Findings

Integral formula involving squared linking numbers

Connection between linking numbers and Conway polynomial coefficients

Extension of modulo 2 results to integral invariants

Abstract

For every spatial embedding of each graph in the Petersen family, it is known that the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2. In this paper, we give an integral lift of this formula in terms of the square of the linking number and the second coefficient of the Conway polynomial.