I am reading the following paper: G. A. Jones, and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), no. 2, 273–307 (MR0505721, Zbl 0391.05024). The authors are interested in imbeddings of an allowed graph $(\mathcal{G},\mathcal{V})$ into a connected, oriented surface $\mathcal{S}$ without boundary; i.e.,

there is a homeomorphism of $\mathcal{G}$ with a subspace of $\mathcal{S}$ [and] we will identify $\mathcal{G}$ with its image in $\mathcal{S}$ …

Here, an "allowed graph" is slightly more general than an undirected pseudograph in that it allows the existence of edges that are free at one end; e.g., let $\mathcal{G} = [0,2] \subseteq \mathbf{R}$, and let $\mathcal{V} = \Set{0,1}$. Then, $\mathcal{G}$ has two edges, $e_1 = [0,1]$ and $e_2 = [1,2]$, the latter being a free edge. We assume that allowed graphs are locally finite.

Now, on page 277, the authors make an additional assumption about the imbeddings of allowed graphs into oriented surfaces:

TM1: whenever $p \in \mathcal{G}$ has valency $k \in \mathbb{N}$, then there is a neighborhood $N_p$ of $p$ in $\mathcal{S}$ and a homeomorphism $\varphi_p \colon N_p \to D = \{ z \in \mathbf{C} \,|\, \lvert z \rvert < 1 \}$ such that $\varphi_p(p) = 0$ and $\varphi_p(N_p \cap \mathcal{G}) = \{ z \in \mathbf{C} \,|\, z^k \in [0,1) \subseteq \mathbf{R} \}$.

This condition is imposed to avoid pathological imbeddings.

Question: What is an example of a pathological imbedding of an allowed graph, i.e., one that violates the assumption TM1?

posted 8 months ago

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8mo ago

The Amplitwist‭

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