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Reidemeister showed that just three basic moves (which manipulated crossings) were needed to test whether two knots were equivalent or "ambient isotopic." Suppose we expand the class of topological objects to permit self-intersection, as here:

self-intersecting

The notion of ambient isotopy applies to such objects. What is the smallest class of moves that guarantees one can transform such an object into any other ambient isotopic object?

I suspect the Reidemeister moves need be augmented by fairly trivial moves passing segments above or beneath the intersected crossing, but how would one prove that?

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These objects are called spatial graphs. A set of Reidemeister moves for them was found by Kauffman; see

L H Kauffman, Invariants of graphs in three-space, Trans. Amer. Math. Soc. 311 (1989) 697–710.

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    $\begingroup$ Perfect. Thanks so much for the reference. (accept) . Indeed, the additional moves are fairly straightforward (including the "twist" of segments at an intersection). $\endgroup$ – David G. Stork Dec 28 '18 at 19:15

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