[Paper Review] Fatgraph Algorithms and the Homology of the Kontsevich Complex

Riccardo Murri|arXiv (Cornell University)|Feb 8, 2012

Algebraic Geometry and Number Theory13 references3 citations

TL;DR

This paper presents efficient algorithms for generating fatgraphs of given genus $g$ and boundary count $n$, computing their automorphisms, and determining the rational homology of the Kontsevich fatgraph complex. By leveraging topological representations and optimized matrix computation, the authors compute Betti numbers of the moduli space $\mathcal{M}_{g,n}$ for $(2g+n)\leq 6$, confirming known results through effective computer implementation.

ABSTRACT

Fatgraphs are multigraphs enriched with a cyclic order of the edges incident to a vertex. This paper presents algorithms to: (1) generate the set of all fatgraphs having a given genus and number of boundary cycles; (2) compute automorphisms of any given fatgraph; (3) compute the homology of the fatgraph complex. The algorithms are suitable for effective computer implementation. In particular, this allows us to compute the rational homology of the moduli space of Riemann surfaces with marked points. We thus compute the Betti numbers of $M_{g,n}$ with $(2g + n) \leq 6$, corroborating known results.

Motivation & Objective

• To develop computationally effective algorithms for enumerating fatgraphs of specified genus $g$ and number of boundary components $n$.
• To compute the automorphism groups of individual fatgraphs using topological and combinatorial representations.
• To determine the rational homology of the fatgraph complex $\mathcal{R}_{g,n}$, enabling computation of Betti numbers of the moduli space $\mathcal{M}_{g,n}$.

Proposed method

• Represent fatgraphs via topological data structures based on cyclic edge orders at vertices, enabling direct manipulation of their surface realizations.
• Use recursive construction from trivalent graphs in $\mathcal{M}_{g-1,n}$ and $\mathcal{M}_{g-1,n+1}$ to generate all $\mathcal{R}_{g,n}$ fatgraphs.
• Implement edge contraction as a differential $D$ mapping $p$-edge fatgraphs to $(p-1)$-edge graphs, represented as sparse matrices $\mathbf{D}^{(p)}$.
• Optimize matrix construction by applying edge contraction to unmarked topological fatgraphs first, reducing complexity by a factor of $O((n!)^2)$ via automorphism group action.
• Extend the resulting matrix to marked fatgraphs by accounting for boundary cycle permutations using automorphism group actions.
• Compute Betti numbers via standard linear algebra: $\text{Betti}_p = \dim W_p - \text{rank}(\mathbf{D}^{(p)})$.

Experimental results

Research questions

• RQ1 How can fatgraphs of genus $g$ and $n$ boundary components be effectively enumerated for computational homology computation?
• RQ2 What is an efficient algorithm to compute the automorphism group of a given fatgraph using its topological structure?
• RQ3 How can the rational homology of the Kontsevich fatgraph complex $\mathcal{R}_{g,n}$ be computed algorithmically?
• RQ4 What optimizations can be applied to the matrix representation of the differential $D$ to reduce computational complexity?
• RQ5 Can the proposed algorithms compute the Betti numbers of $\mathcal{M}_{g,n}$ for small $g$ and $n$ with high confidence?

Key findings

• The authors successfully compute the rational Betti numbers of the moduli space $\mathcal{M}_{g,n}$ for all pairs with $2g + n \leq 6$, confirming known results.
• The algorithm for generating $\mathcal{R}_{g,n}$ is based on recursive trivalent graph extension, ensuring completeness and correctness for the given genus and boundary count.
• By applying edge contraction to unmarked topological fatgraphs first, the method reduces the complexity of matrix construction by a factor of $O((n!)^2)$ compared to naive approaches.
• The differential $D$ is implemented as a matrix $\mathbf{D}^{(p)}$ mapping $p$-edge fatgraph bases to $(p-1)$-edge bases, with orientation signs computed via topological consistency.
• The final Betti numbers are derived from the rank of $\mathbf{D}^{(p)}$ and the dimension of chain spaces $W_p$, using standard linear algebra.
• The implementation is effective and efficient, with a working Python-based prototype demonstrating feasibility for small $g$ and $n$.