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Flow polynomials as Feynman amplitudes and their $alpha$-representation | | Eduard Yu. Lerner
; Andrey P. Kuptsov
; Sofya A. Mukhamedjanova
; | | Date: |
5 Sep 2016 | | Abstract: | Let $G$ be a connected graph; denote by $ au(G)$ the set of its spanning
trees. Let $mathbb F_q$ be a finite field, $s(alpha,G)=sum_{Tin au(G)}
prod_{e in E(T)} alpha_e$, where ${alpha_ein mathbb F_q}$. Kontsevich
conjectured in 1997 that the number of nonzero values of $s(alpha, G)$ is a
polynomial in $q$ for all graphs. This conjecture was disproved by Brosnan and
Belkale. In this paper, using the standard technique of the Fourier
transformation of Feynman amplitudes, we express the flow polynomial $F_G(q)$
in terms of the "correct" Kontsevich formula. Our formula represents $F_G(q)$
as a linear combination of Legendre symbols of $s(alpha, H)$ with coefficients
$pm 1/q^{(|V(H)|-1)/2}$, where $H$ is a contracted graph of $G$ depending on
$alphain left(mathbb F^*_q
ight)^{E(G)}$, and $|V(H)|$ is odd. The case
$q=5$ corresponds to the least number with which all coefficients in the linear
combination are positive. This allows us to hope that the obtained result can
be applied to prove the Tutte 5-flow conjecture. | | Source: | arXiv, 1609.1120 | | Services: | Forum | Review | PDF | Favorites | |
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