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 $\mathbf{R}^2$. Morphisms $A\to B$ are embeddings of unions of finitely many closed intervals and circles into $[0,1]\times\mathbf{R}^2$ such that the restriction of the embedding to the endpoints yields a bijection to $A\sqcup B$.

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