Communications in Number Theory and Physics

Volume 17 (2023)

Number 2

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

Pages: 343 – 384



Simone Hu (Mathematical Institute, University of Oxford, United Kingdom)

Karen Yeats (Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Ontario, Canada)


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.


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