nLab
tangle

Contents

Contents

Idea

A tangle is like a knot that was cut at several points and the resulting strands were pulled apart.

Definition

Tangles form a category. Its objects are finite subsets of R 2\mathbf{R}^2. Morphisms ABA\to B are embeddings of unions of finitely many closed intervals and circles into [0,1]×R 2[0,1]\times\mathbf{R}^2 such that the restriction of the embedding to the endpoints yields a bijection to ABA\sqcup B.

Morphisms are composed by gluing two copies of [0,1][0,1] 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 ABA\to B will encode the whole homotopy type of the space of embeddings described above.

References

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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.