Graph invariants from the topology of rigid isotopy classes

Abstract

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 $π_k(W_{G, d}) = 0$ 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.

Figures (3)

• Figure 1: Here we are drawing points in $\mathbb{R}^2$ together with the circles centered at those points and with radius $\frac{1}{\sqrt{2}}$. Now, let us define $P$ and $P'$ points in $\mathbb{R}^{2\times 25}$ in such a way that $p_1$ is the point inside the big circle and $p_1'$ is the point outside the big circle while $p_i=p_i'$ for $i>1$ and they are the points on the big circle. Then, the two geometric graphs $G(P)$ and $G(P')$ are isomorphic but not rigidly isotopic.
• Figure 2: A plot of the Betti numbers of $W_{G, d}$. The width of each non zero cluster of holes is $\binom{n}{2}+1$, which is a constant. Each of these clusters is placed at a multiple of $d$ and as $d\to \infty$ they shift to infinity. The total Betti number of $W_{G, d}$, i.e. the blue area, becomes constant for $d$ large enough.
• Figure 3: This is a schematic image of the $E_2(\epsilon)$ term of the spectral sequence we are describing. The coloured parts correspond to the elements $E_2^{i,j}(\epsilon)$ of the spectral sequence which are possibly non-zero.

Theorems & Definitions (91)

• Definition 1
• Theorem 2
• Example 3: Homotopy groups of the configuration space of $n$ points in $\mathbb{R}^d$
• Theorem 4
• Example 5: The infinite dimensional sphere
• Example 6: The iterated suspension
• Theorem 7
• Example 8: Betti numbers of the configuration space of $n$ points in $\mathbb{R}^d$
• Theorem 9
• Theorem 10