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Let $\Sigma$ be a compact oriented surface with boundary and $L = \mathrm{Imm}(\bigsqcup_{i=1}^n S^1,\Sigma)$ the space of all generic (i.e. transversally and at most doubly intersecting) immersions of $n$ loops. By applying a sequence of elementary moves (which are 2D analogue of Reidemeister moves), we can traverse $\pi_0(L)$, the set of connected components of $L$ w.r.t. the compact-open topology.

Now we can draw a graph such that the set of vertices is exactly $\pi_0(L)$ and there is an edge between vertices iff they can be connected by a single application of an elementary move.

Does this graph have a name? Are there any papers/references about this topic?

Any comments are appreciated.

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    $\begingroup$ There's a variety of names ascribed to these kinds of graphs. Some of the earlier occurrences of this construction is in the work of Arnold, Vassiliev and Cerf. In general if you have a stratified space you can form a graph of the components of the interior of the top-dimensional strata, with the edges the co-dimension 1 strata. For stratified spaces this is likely just called the dual graph to the co-dimension 1 stratum, but you might want to check a reference like Goresky-Macpherson. $\endgroup$
    – Ryan Budney
    Commented Oct 10, 2023 at 16:50
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    $\begingroup$ Maybe "adjacency graph" is one of the more helpful terms to readers. In different areas it gets different names. For knots in $\mathbb R^3$ using crossing changes it is called the "Gordian graph". $\endgroup$
    – Ryan Budney
    Commented Oct 10, 2023 at 17:11

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