Completing the $c_2$ completion conjecture for $p=2$

TL;DR

This paper proves the completion invariance of the $c_2$-invariant for $p=2$, a key step in understanding Feynman periods, using combinatorial and enumerative methods.

Contribution

It establishes the $p=2$ case of the $c_2$-invariant completion conjecture, extending prior results and employing novel combinatorial techniques.

Findings

Proves completion invariance of $c_2$-invariant at $p=2$

Extends previous work on $c_2$-invariant symmetry

Uses combinatorial and enumerative methods for proof

Abstract

The $c_2$-invariant is an arithmetic graph invariant useful for understanding Feynman periods. Brown and Schnetz conjectured that the $c_2$-invariant has a particular symmetry known as completion invariance. This paper will prove completion invariance of the $c_2$-invariant in the $p=2$ case, extending previous work of one of us. The methods are combinatorial and enumerative involving counting certain partitions of the edges of the graph.