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

Abstract

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.

23 pages, 7 figures; revised and extended version

References in corpus (5)

• Generation of Matrix Models by W-operators
• The spectral curve of the Eynard-Orantin recursion via the Laplace transform
• McGenus: A Monte Carlo algorithm to predict RNA secondary structures with pseudoknots
• Topological recursion for chord diagrams, RNA complexes, and cells in moduli spaces
• Some formulas for the number of gluings

Cited by in corpus (5)

• Combinatorial solutions to integrable hierarchies
• Partial chord diagrams and matrix models
• Generalized Hultman Numbers and Cycle Structures of Breakpoint Graphs
• The boundary length and point spectrum enumeration of partial chord diagrams using cut and join recursion
• On gluing a surface of genus $g$ from two and three polygons