A tangle is like a knot that was cut at several points and the resulting strands were pulled apart.
Tangles form a category. Its objects are finite subsets of . Morphisms are embeddings of unions of finitely many closed intervals and circles into such that the restriction of the embedding to the endpoints yields a bijection to .
Morphisms are composed by gluing two copies of together and rescaling.
As usual, this suffers from being associative only up to an ambient isotopy. Thus, one can either take ambient isotopy classes of such embeddings, obtaining a 1-category of tangles, or instead turn tangles into an (∞,1)-category, in which case morphisms will encode the whole homotopy type of the space of embeddings described above.
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Last revised on April 6, 2020 at 03:21:21. See the history of this page for a list of all contributions to it.