A short proof of the generalized Conway--Gordon--Sachs theorem

TL;DR

This paper presents a concise proof of the generalized Conway--Gordon--Sachs theorem for complete graphs, simplifying previous complex proofs and extending the theorem over integers.

Contribution

A shorter, more straightforward proof of the generalized Conway--Gordon--Sachs theorem over integers is provided, improving upon prior lengthy proofs.

Findings

Shorter proof of the theorem over integers

Extension of the theorem to complete graphs on n vertices

Simplification of previous complex proofs

Abstract

The famous Conway--Gordon--Sachs theorem for the complete graph on six vertices was extended to the general complete graph on $n$ vertices by Kazakov--Korablev as a congruence modulo $2$, and its integral lift was given by Morishita--Nikkuni. However, the proof is complicated and long. In this paper, we provide a shorter proof of the generalized Conway--Gordon--Sachs theorem over integers.