Skip to main content Accessibility help

A Special Case of Completion Invariance for the c 2 Invariant of a Graph

  • Karen Yeats (a1)


The ${{c}_{2}}$ invariant is an arithmetic graph invariant defined by Schnetz. It is 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 case where we are over the field with 2 elements and the completed graph has an odd number of vertices. The methods involve enumerating certain edge bipartitions of graphs; two different constructions are needed.



Hide All
[1] Ax, J., Zeroes of polynomials over finite fields. Amer. J. Math. 86 (1964), 255261. http://dx.doi.Org/10.2307/2373163
[2] Belkale, P. and Brosnan, P., Matroids, motives, and a conjecture of Kontsevich. Duke Math. J. 116 (2003), 147188.
[3] Bloch, S., Esnault, H., and Kreimer, D., On motives associated to graph polynomials. Comm. Math. Phys. 267 (2006), 181225. http://dx.doi.Org/10.1007/s00220-006-0040-2
[4] Broadhurst, D. and Schnetz, O., Algebraic geometry informs perturbative quantum field theory. In PoS, volume LL2014, page 078, 2014. arxiv:1 409.5570
[5] Broadhurst, D. J. and Kreimer, D., Knots and numbers in ϕ4 theory to 7 loops and beyond. Internat. J. Modern Phys. C 6 (1995), 519524. http://dx.doi.Org/10.1142/S012918319500037X
[6] Brown, F., Feynman amplitudes, coaction principle, and cosmic Galois group. Commun. Number Theory Phys. 11 (2017), no. 3, 453556. http://dx.doi.Org/10.4310/CNTP.2017.v11.n3.a1
[7] Brown, F., On the periods of some Feynman integrals. arxiv:0910.0114
[8] Brown, F. and Schnetz, O., A K3 in ϕ4. Duke Math J. 161 (2012), 18171862.
[9] Brown, F. and Schnetz, O., Modular forms in quantum field theory. Commun. Number Theory Phys. 7 (2013), 293325.
[10] Brown, F., Schnetz, O., and Yeats, K., Properties ofC2 invariants of Feynman graphs. Adv. Theor. Math. Phys. 18 (2014), 323362.
[11] Brown, F. and Yeats, K., Spanning forest polynomials and the transcendental weight of Feynman graphs. Comm. Math. Phys. 301 (2011), 357382.
[12] Chaiken, S., A combinatorial proof of the all minors matrix tree theorem. SIAM J. Alg. Disc. Meth. 3 (1982), 319329. http://dx.doi.Org/10.1137/0603033
[13] Chorney, W. and Yeats, K., C2 invariants of recursive families of graphs. arxiv:1701.01208
[14] Crump, I., Graph Invariants with Connections to the Feynman Period in 4 Theory. PhD thesis, Simon Fraser University, 2017. arxiv:1704.06350
[15] Cummins, R. L., Hamilton circuits in tree graphs. IEEE Trans. Circuit Theory CT-13 (1966), 8290. http://dx.doi.Org/1 0.1109/TCT.1 966.1082546
[16] Diestel, R., Graph theory. Graduate Texts in Mathematics, 173, Springer-Verlag, New York, 1997.
[17] Doryn, D., The C2 invariant is invariant. arxiv:1312.7271
[18] Doryn, D., Dual graph polynomials and a i-face formula. arxiv:1508.03484
[19] Itzykson, C. and Zuber, J.-B., Quantum field theory. International Series in Pure and Applied Physics, McGraw-Hill, 1980. Dover edition 2005.
[20] Logan, A., New realizations of modular forms in Calabi-Yau threefolds arising from ϕ4 theory. arxiv:1 604.0491 8
[21] Marcolli, M., Feynman motives. World Scientific, Hackensack, NJ, 2010.
[22] Schnetz, O., Numbers and functions in quantum field theory. arxiv:1606.08598
[23] Schnetz, O., Quantum periods: A census of ϕ4 -transcendentals. Commun. Number Theory Phys. 4 (2010), 147. .a1
[24] Schnetz, O., Quantum field theory over Fq. Electron. J. Combin. 18 (2011), no. 1, Paper 102.
[25] Vlasev, A. and Yeats, K., A four-vertex, quadratic, spanning forest polynomial identity. Electron. J. Linear Algebra 23 (2012), 923941. http://dx.doi.Org/10.13001/1081-3810.1566
[26] Yeats, K., A few c2 invariants of circulant graphs. Commun. Number Theory Phys. 10 (2016), 6386. .a3
MathJax is a JavaScript display engine for mathematics. For more information see


Related content

Powered by UNSILO

A Special Case of Completion Invariance for the c 2 Invariant of a Graph

  • Karen Yeats (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.