A tangle is a 1-manifoldMathworldPlanetmath, i.e. a disjoint unionMathworldPlanetmathPlanetmath of arcs and circles, embedded in (0,1)2×[0,1]. The boundary of a tangle is contained in (0,1)2×{0,1}. Two tangles are considered equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if and only if they are ambient isotopic relative to their boundaries. Combinatorially, tangles can be understood as tangle diagrams. Any two tangle diagrams which represent the same tangle can be connectedPlanetmathPlanetmath by Reidemeister movesMathworldPlanetmath. This is the content of a slight generalizationPlanetmathPlanetmath of Reidemeister’s theorem. Algebraically, tangles form the morphismsMathworldPlanetmathPlanetmath of a tortile monoidal category. This is a corollary of Shum’s theorem. Specifically, they form the tortile monoidal category generated by a self-dual,unframed object.

Title tangle
Canonical name Tangle
Date of creation 2013-03-22 18:16:41
Last modified on 2013-03-22 18:16:41
Owner apollonius (16438)
Last modified by apollonius (16438)
Numerical id 5
Author apollonius (16438)
Entry type Definition
Classification msc 54C25
Related topic Knot
Related topic Link
Related topic Braid