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Enumeration of chord diagrams on many intervals and their non-orientable analogs

Alexeev, N. V. and Andersen, J. E. and Penner, R. C. and Zograf, P. G. (2016) Enumeration of chord diagrams on many intervals and their non-orientable analogs. Advances in Mathematics, 289 . pp. 1056-1081. ISSN 0001-8708. doi:10.1016/j.aim.2015.11.032.

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Two types of connected chord diagrams with chord endpoints lying in a collection of ordered and oriented real segments are considered here: the real segments may contain additional bivalent vertices in one model but not in the other. In the former case, we record in a generating function the number of fatgraph boundary cycles containing a fixed number of bivalent vertices while in the latter, we instead record the number of boundary cycles of each fixed length. Second order, non-linear, algebraic partial differential equations are derived which are satisfied by these generating functions in each case giving efficient enumerative schemes. Moreover, these generating functions provide multi-parameter families of solutions to the KP hierarchy. For each model, there is furthermore a non-orientable analog, and each such model likewise has its own associated differential equation. The enumerative problems we solve are interpreted in terms of certain polygon gluings. As specific applications, we discuss models of several interacting RNA molecules. We also study a matrix integral which computes numbers of chord diagrams in both orientable and non-orientable cases in the model with bivalent vertices, and the large-N limit is computed using techniques of free probability.

Item Type:Article
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URLURL TypeDescription Paper
Andersen, J. E.0000-0001-9721-0722
Additional Information:© 2015 Elsevier Inc. Research supported by the Centre for Quantum Geometry of Moduli Spaces which is funded by the Danish National Research Foundation. The work of N.A. and P.Z. was further supported by the Government of the Russian Federation megagrant 11.G34.31.0026, by JSC “Gazprom Neft”, and by the RFBR grants 13-01-00935-a and 13-01-12422-OFI-M2. In addition, N.A. was partially supported by the RFBR grant 14-01-00500-a and by the SPbSU grant 6.38.672.2013, and P.Z. was partially supported by the RFBR grant 14-01-00373-a.
Funding AgencyGrant Number
Danish National Research FoundationUNSPECIFIED
Russian Federation11.G34.31.0026
Russian Foundation for Basic Research (RFBR)14-01-00500-a
Saint Petersburg State University (SPbSU)6.38.672.2013
Russian Foundation for Basic Research (RFBR)14-01-00373-a
Subject Keywords:Chord diagrams; Polygon gluings; Random matrices
Record Number:CaltechAUTHORS:20160218-150306233
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Official Citation:N.V. Alexeev, J.E. Andersen, R.C. Penner, P.G. Zograf, Enumeration of chord diagrams on many intervals and their non-orientable analogs, Advances in Mathematics, Volume 289, 5 February 2016, Pages 1056-1081, ISSN 0001-8708, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:64590
Deposited By: Tony Diaz
Deposited On:18 Feb 2016 23:17
Last Modified:10 Nov 2021 23:33

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