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# Communications in Number Theory and Physics

## Volume 17 (2023)

### Number 2

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

Pages: 343 – 384

DOI: https://dx.doi.org/10.4310/CNTP.2023.v17.n2.a4

#### Authors

#### 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.

#### Keywords

Feynman period, completion, $c_2$-invariant, edge partition

#### 2010 Mathematics Subject Classification

Primary 81T18. Secondary 05C30, 05C31, 81Q30.

K.Y. is supported by an NSERC Discovery grant and by the Canada Research Chairs program.

Received 23 June 2022

Accepted 22 March 2023

Published 4 May 2023