A study on prefixes of $c_2$ invariants

TL;DR

This paper reviews recent combinatorial approaches to the $c_2$ invariant, reports new calculations for circulant graphs, and suggests all finite sequences can appear as initial segments of these invariants.

Contribution

It introduces new calculations of $c_2$ invariants for circulant graphs and explores the diversity of initial segments, challenging previous assumptions about their sparsity.

Findings

All finite sequences appear as initial segments of $c_2$ invariants.

Recent calculations support the combinatorial perspective.

The study links $c_2$ invariants to Feynman integrals.

Abstract

This document begins by reviewing recent progress that has been made by taking a combinatorial perspective on the $c_2$ invariant, an arithmetic graph invariant with connections to Feynman integrals. Then it proceeds to report on some recent calculations of $c_2$ invariants for two families of circulant graphs at small primes. These calculations support the idea that all possible finite sequences appear as initial segments of $c_2$ invariants, in contrast to their apparent sparsity on small graphs.