Form of presentation | Articles in international journals and collections |
Year of publication | 2017 |
Язык | английский |
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Lerner Eduard Yulevich, author
Mukhamedzhanova Sofya Alfisovna, author
|
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Kupcov Andrey Pavlovich, author
|
Bibliographic description in the original language |
A. P. Kuptsov, E. Yu. Lerner, S. A. Mukhamedjanova, Flow polynomials as Feynman amplitudes and their $\alpha$-representation, Electron. J. Combin., 24 (2017), no. 1, paper 11, 19 pp. |
Annotation |
Let $G$ be a connected graph; denote by $\tau(G)$ the set of its spanning trees. Let $\mathbb F_q$ be a finite field, $s(\alpha,G)=\sum_{T\in\tau(G)} \prod_{e \in E(T)} \alpha_e$, where $\alpha_e\in \mathbb F_q$. 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 $\alpha\in \left(\mathbb F^*_q\right)^{E(G)}$, and $|V(H)|$ is odd. |
Keywords |
Flow polynomial, Kontsevich's conjecture, Laplacian matrix, Feynman amplitudes, Legendre symbol, Tutte 5-flow conjecture |
The name of the journal |
The Electronic Journal of Combinatorics
|
Please use this ID to quote from or refer to the card |
https://repository.kpfu.ru/eng/?p_id=234426&p_lang=2 |
Full metadata record ![](https://shelly.kpfu.ru/pdf/picture/arrow_black_right.gif) |
Field DC |
Value |
Language |
dc.contributor.author |
Lerner Eduard Yulevich |
ru_RU |
dc.contributor.author |
Mukhamedzhanova Sofya Alfisovna |
ru_RU |
dc.contributor.author |
Kupcov Andrey Pavlovich |
ru_RU |
dc.date.accessioned |
2017-01-01T00:00:00Z |
ru_RU |
dc.date.available |
2017-01-01T00:00:00Z |
ru_RU |
dc.date.issued |
2017 |
ru_RU |
dc.identifier.citation |
A. P. Kuptsov, E. Yu. Lerner, S. A. Mukhamedjanova, Flow polynomials as Feynman amplitudes and their $\alpha$-representation, Electron. J. Combin., 24 (2017), no. 1, paper 11, 19 pp. |
ru_RU |
dc.identifier.uri |
https://repository.kpfu.ru/eng/?p_id=234426&p_lang=2 |
ru_RU |
dc.description.abstract |
The Electronic Journal of Combinatorics |
ru_RU |
dc.description.abstract |
Let $G$ be a connected graph; denote by $\tau(G)$ the set of its spanning trees. Let $\mathbb F_q$ be a finite field, $s(\alpha,G)=\sum_{T\in\tau(G)} \prod_{e \in E(T)} \alpha_e$, where $\alpha_e\in \mathbb F_q$. 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 $\alpha\in \left(\mathbb F^*_q\right)^{E(G)}$, and $|V(H)|$ is odd. |
ru_RU |
dc.language.iso |
ru |
ru_RU |
dc.subject |
Flow polynomial |
ru_RU |
dc.subject |
Kontsevich's conjecture |
ru_RU |
dc.subject |
Laplacian matrix |
ru_RU |
dc.subject |
Feynman amplitudes |
ru_RU |
dc.subject |
Legendre symbol |
ru_RU |
dc.subject |
Tutte 5-flow conjecture |
ru_RU |
dc.title |
Flow polynomials as Feynman amplitudes and their $\alpha$-representation |
ru_RU |
dc.type |
Articles in international journals and collections |
ru_RU |
|