# Journal of Nonlinear Mathematical Physics

Volume 24, Issue Supplement 1, December 2017, Pages 157 - 173

# The heptagon-wheel cocycle in the Kontsevich graph complex

Authors
Ricardo Buring
Institut für Mathematik, Johannes Gutenberg–Universität, Staudingerweg 9, D-55128 Mainz, Germany.rburing@uni-mainz.de
Arthemy V. Kiselev
Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands,A.V.Kiselev@rug.nl
Nina J. Rutten
Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands
Received 2 October 2017, Accepted 24 November 2017, Available Online 6 January 2021.
DOI
10.1080/14029251.2017.1418060How to use a DOI?
Keywords
Non-oriented graph complex; differential; cocycle; symmetry; Poisson geometry
Abstract

The real vector space of non-oriented graphs is known to carry a differential graded Lie algebra structure. Cocycles in the Kontsevich graph complex, expressed using formal sums of graphs on n vertices and 2n − 2 edges, induce – under the orientation mapping – infinitesimal symmetries of classical Poisson structures on arbitrary finite-dimensional affine real manifolds. Willwacher has stated the existence of a nontrivial cocycle that contains the (2ℓ + 1)-wheel graph with a nonzero coefficient at every ℓ∈ℕ. We present detailed calculations of the differential of graphs; for the tetrahedron and pentagon-wheel cocycles, consisting at ℓ = 1 and ℓ = 2 of one and two graphs respectively, the cocycle condition d(γ) = 0 is verified by hand. For the next, heptagonwheel cocycle (known to exist at ℓ = 3), we provide an explicit representative: it consists of 46 graphs on 8 vertices and 14 edges.

Copyright
© 2017 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
24 - Supplement 1
Pages
157 - 173
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2017.1418060How to use a DOI?
Copyright
© 2017 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

### Cite this article

TY  - JOUR
AU  - Ricardo Buring
AU  - Arthemy V. Kiselev
AU  - Nina J. Rutten
PY  - 2021
DA  - 2021/01/06
TI  - The heptagon-wheel cocycle in the Kontsevich graph complex
JO  - Journal of Nonlinear Mathematical Physics
SP  - 157
EP  - 173
VL  - 24
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2017.1418060
DO  - 10.1080/14029251.2017.1418060
ID  - Buring2021
ER  -