This is a comment requesting clarification, not an answer.
At least to me, it is not self-evident what is the 4-valent equivalent of the three Reidemeister
moves. For example, Reidemeister I seems fundamentally 3-valent.
If you allow the move shown below, which is akin to Reidemeister II, and if your collection
of curves includes only proper crossings, as illustrated, then indeed
the graph can be converted to a collection of disjoint (but often nested) loops:
in the example, four loops nested inside a fifth.
Perhaps you could clarify your intent?
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This is a comment, not an answer.
At least to me, it is not self-evident what is the 4-valent equivalent of the three Reidemeister
moves. For example, Reidemeister I seems fundamentally 3-valent.
If you allow the move shown below, which is akin to Reidemeister II, and if your collection
of curves includes only proper crossings, as illustrated, then indeed
the graph can be converted to a collection of disjoint (but often nested) loops:
in the example, four loops nested inside a fifth.
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