nLab tangle

Idea

An ordinary tangle is a 1-dimensional manifold with boundary that is embedded into the 3-cube, such that the boundary is embedded into chosen chosen pair opposing sides of the cube.

Thus, one may think of a tangle as (a knot if it is connected or generally as) a link that was cut at several points and the resulting strands pulled apart at their endpoints to opposite sides of the cube. Conversely, a link (knot) is equivalently a (connected) tangle with empty boundary.

Similarly, a tangle that progresses monotonically from its source to its target is equivalently a braid.

The notion of tangles generalizes to that of $m$-tangles in dimension $n$, which are $m$-manifolds with corners embedded into the $n$-cube such that their corners are appropriately embedded in the cube’s boundary. In this sense ordinary tangles are the 1-tangles in 3-space.

Properties

Category of tangles

Tangles naturally constitute the morphisms of a category:

The 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.

Category of framed tangles

Analogously there is a notion of framed tangles which are to ordinary tangles as framed links are to ordinary links.

Proposition

(Shum's theorem)
The category of framed oriented tangles is equivalently the free ribbon category generated by a single object.

(Shum 1994, Yetter 2001 Thm. 9.1)

Higher-dimensional variants

Higher-dimensional tangles, i.e. $m$-manifolds with corners embedded in the $n$-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.

Related concepts

• generalized tangle hypothesis

• manifold diagrams

References

• Mei Chee Shum, Tortile tensor categories, Journal of Pure and Applied Algebra 93 1 (1994) 57-110 [10.1016/0022-4049(92)00039-T]

• John Baez, James Dolan: Higher-dimensional Algebra and Topological Quantum Field Theory (1995) [arXiv]

• David N. Yetter: Functorial Knot Theory – Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants, Series on Knots and Everything 26, World Scientific (2001) [doi:10.1142/4542]

• Christoph Dorn, Christopher Douglas,: Manifold diagrams and tame tangles (2022) [arXiv:2208.13758]

See also:

• Wikipedia: Tangle (mathematics)

Exposition

• Tangled Web: The category of tangles (2009)