# tangle

A tangle is a $1$-manifold^{}, i.e. a disjoint union^{} of arcs and circles, embedded in ${(0,1)}^{2}\times [0,1]$. The boundary of a tangle is contained in ${(0,1)}^{2}\times \{0,1\}$. Two tangles are considered equivalent^{} 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 connected^{} by Reidemeister moves^{}. This is the content of a slight generalization^{} of Reidemeister’s theorem. Algebraically, tangles form the morphisms^{} 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 |
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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 |