An ordinary tangle is 1-manifold with boundary that is embedded in the 2-cube (or 3-cube), such that the tangle’s boundary is embedded in two chosen opposing sides of the cube. Thus, one may think of a tangle as a knot that was cut at several points and the resulting strands were pulled apart at their endpoints to opposite sides of the cube.

The notion generalizes to that of $m$-tangles in dimension $n$, which are $m$-manifolds with corners embedded in the $n$-cube that, such that their corners are appropriately embedded in the $n$-cube’s boundary.

Definition of ordinary tangles

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.

Higher-dimensional variants

Higher-dimensional tangles, i.e. $m$-manifold with corners embedded in the $n$-cube, were considered for instance in Baez and Dolan 95. A “tame” definition of tangles that admit finitestratifications by their critical point types was given in Dorn and Douglas 22.