论文标题
刚性同位素类别拓扑的图形不变性
Graph invariants from the topology of rigid isotopy classes
论文作者
论文摘要
我们使用一种让人联想到浮雕同源性的限制程序来定义一个新的图形不变性家族，研究其在欧几里得空间中几何实现的模量空间的拓扑。给定$ n $顶点上的标记图$ g $，$ d \ geq 1 $，$ w_ {g，d} \ subseteq \ subseteq \ subseteq \ mathbb {r}^{d \ times n} $表示非eNondepections $ g $的空间它是非空的，我们将其连接的组件称为$ g $ in $ \ m mathbb {r}^d $的刚性同位素类别。我们研究了这些刚性同位素类别的拓扑。 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.虽然$π_k（w_ {g，d}）= 0 $ for $ g $，固定和$ d $足够大，但我们还证明，尽管如此，当$ d \ to \ infty $ $ w_ {g，d} $的不变同源性结构时$ d $，对于$ d $，足够大。这导致了一个图形不变的家族的定义，从而捕获了这种结构。例如，$ w_ {g，d} $的betti数字的总和不取决于$ d $，对于$ d $，足够大；我们称此号码为图$ g $的浮数。最后，我们对$ \ mathbb {r}^d $的刚性同位素类别的数量进行渐近估计 - $ n $ VERTICES上的$ n $ Vertices for $ d $固定和$ n $ tody to Infinity的几何图。当$ d = 1 $时，我们表明，每个同构类别类别均为$ n \ to \ infty $，平均与恒定数量的刚性同位素类别相对应。对于$ d> 1 $，我们在对数尺度上证明了类似的声明。
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.
