nLab tangle

Contents

Contents

Idea

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 mm-tangles in dimension nn, which are mm-manifolds with corners embedded in the nn-cube that, such that their corners are appropriately embedded in the nn-cube’s boundary.

Definition of ordinary tangles

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.

Higher-dimensional variants

Higher-dimensional tangles, i.e. mm-manifold with corners embedded in the nn-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.

References

  • 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 (pdfs)

Last revised on September 23, 2022 at 13:49:27. See the history of this page for a list of all contributions to it.