Graph invariants from the topology of rigid isotopy classes

• Published in: arXiv.org
• Main Authors: Belotti, Mara, Lerario, Antonio, Newman, Andrew
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 10.08.2020
• Subjects: Apexes; Asymptotic properties; Euclidean geometry; Graph theory; Homology; Invariants; Isomorphism; Mathematics - Algebraic Geometry; Mathematics - Algebraic Topology; Topology
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2008.03984

We define a new family of graph invariants, studying the topology of the moduli space of their geometric realizations in Euclidean spaces, using a limiting procedure reminiscent of Floer homology. Given a labeled graph \(G\) on \(n\) vertices and \(d \geq 1\), \(W_{G, d} \subseteq \mathbb{R}^{d \times n}\) denotes the space of nondegenerate realizations of \(G\) in \(\mathbb{R}^d\).The set \(W_{G, d}\) might not be connected, even when it is nonempty, and we refer to its connected components as rigid isotopy classes of \(G\) in \(\mathbb{R}^d\). We study the topology of these rigid isotopy classes. First, regarding the connectivity of \(W_{G, d}\), we generalize a result of Maehara that \(W_{G, d}\) is nonempty for \(d \geq n\) to show that \(W_{G, d}\) is \(k\)-connected for \(d \geq n + k + 1\), and so \(W_{G, \infty}\) is always contractible. While \(\pi_k(W_{G, d}) = 0\) for \(G\), \(k\) fixed and \(d\) large enough, we also prove that, in spite of this, when \(d\to \infty\) the structure of the nonvanishing homology of \(W_{G, d}\) exhibits a stabilization phenomenon: it consists of \((n-1)\) equally spaced clusters whose shape does not depend on \(d\), for \(d\) large enough. This leads to the definition of a family of graph invariants, capturing this structure. For instance, the sum of the Betti numbers of \(W_{G,d}\) does not depend on \(d\), for \(d\) large enough; we call this number the Floer number of the graph \(G\). Finally, we give asymptotic estimates on the number of rigid isotopy classes of \(\mathbb{R}^d\)--geometric graphs on \(n\) vertices for \(d\) fixed and \(n\) tending to infinity. When \(d=1\) we show that asymptotically as \(n\to \infty\) each isomorphism class corresponds to a constant number of rigid isotopy classes, on average. For \(d>1\) we prove a similar statement at the logarithmic scale.