I am currently trying to understand the paper Total positivity, Grassmannians, and networks. In this paper a plabic graphs $G$ is defined as planar graph drawn inside a disk and considered "up to homotopy". On the boundary of that disk lie $n$ vertices labeled $b_1, ... , b_n$ in clockwise order. All other vertices are strictly inside the disk, and colerd black or white. Multiple edges as well as loops are allowed, and all $b_i$ are requiered to be incident with exactly one internal vertex.
This may already be wrong, but i interpret the boundary of the disk belonging to $G$. That means, viewed as a graph in the plane, the outer face of $G$ is bounded by the cycle $b_1b_2...b_nb_1$.
Now i have multiple Questions:
- What should a homotopy be in this context ? Is it a homotopy $f_t:G \to \mathbb{R}^2, t \in [0,1]$, such that all $f_t$ are homeomorphisms onto their image (the concept of homotopy is new to me, but i think this is usually called an isotopy) and $f_0$ is just the inclusion ?
- Suppose 1. is correct, can such an isotopy be covered by an isotopy $F_t: \mathbb{R}^2 \to \mathbb{R}^2$ in the sense that ${F_t}_{|G} = f_t$ for all $t$ and $F_0 = id$ ? Are there nessesary or sufficient conditions ?
To formulate my last Question i should first give some background:
We can order the edges around a vertex $v$ of $G$ clockwise (cf. Graphs on Surfaces chapter 3.2 [sadly i couldnt find a link to a pdf...]): Put a simple closed polygonal curve $C$ around $v$ that intersects all its edges. Every such edge has a first intersection with $C$ when we pass through it starting at $v$. It turns out that the clockwise order of those intersection points does not depend on $C$, so we can order the edges around $v$ acordingly. Such an ordering is usually called the rotation around $v$.
An isotopy as in 1. should preserve rotations. In fact after making some adjustments (we dont assume $G$ to be piecewise linear), the proof given in Simplest proof that cyclic ordering of edges is preserved under planar graph homotopy? should hold.
One can also look at what a homeomorphism $\psi:\mathbb{R}^2 \to \mathbb{R}^2$ can do to such an rotation. Here is what i came up with: Consider a plane $k$-star $S$ (a graph consisting of a central vertex, with $k$ leaves), and assume $k \geq 3$. Then the rotation around the center $z$ of $S$ can at worst be reversed by $\psi$ (here we identify $S$ with its image under $\psi$). Now if $\psi$ preserves the rotation of $z$, then it will also preserve the rotation around the center of any other plane $l$-star, where $l$ is arbitrary. We can thus speak of homeomorphisms of the plane that preserve rotations.
- If 2. is true, then $F_1$ should be a homeomorphism of the plane that preserves rotations (Note that $G$ does contain vertices of degree at least $3$ and we can view them as $k$-stars). Conversely, if we start with such an homeomorphism $F_1$, is there an isotopy $F_t:\mathbb{R}^2 \to \mathbb{R}^2, t \in [0,1]$, with $F_0 = id$ ?