Conway-Gordon type theorem for the complete four-partite graph $K_{3,3,1,1}$

TL;DR

This paper establishes a Conway-Gordon type formula for the complete four-partite graph K_{3,3,1,1}, linking knot invariants with graph structure, and proves the existence of nontrivial Hamiltonian knots in rectilinear embeddings.

Contribution

It introduces a new Conway-Gordon type formula for K_{3,3,1,1} and demonstrates that all rectilinear spatial embeddings contain a nontrivial Hamiltonian knot.

Findings

A Conway-Gordon type formula for K_{3,3,1,1}

Every rectilinear spatial K_{3,3,1,1} contains a nontrivial Hamiltonian knot

Linking number and Conway polynomial coefficients are related in the formula

Abstract

We give a Conway-Gordon type formula for invariants of knots and links in a spatial complete four-partite graph $K_{3,3,1,1}$ in terms of the square of the linking number and the second coefficient of the Conway polynomial. As an application, we show that every rectilinear spatial $K_{3,3,1,1}$ contains a nontrivial Hamiltonian knot.