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 -tangles in dimension , which are -manifolds with corners embedded in the -cube that, such that their corners are appropriately embedded in the -cube’s boundary.
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.
Higher-dimensional tangles, i.e. -manifold with corners embedded in the -cube, were considered for instance in Baez and Dolan 95. A “tame” definition of tangles that admit finite stratifications by their critical point types was given in Dorn and Douglas 22.
John Baez and James Dolan, Higher-dimensional Algebra and Topological Quantum Field Theory 1995 (arXiv)
Christoph Dorn and Christopher Douglas, Manifold diagrams and tame tangles, 2022 (arXiv)
Last revised on March 9, 2023 at 12:03:03. See the history of this page for a list of all contributions to it.