# connected sum

The *connected sum ^{} of knots* $K$ and $J$ is a knot, denoted by $K\mathrm{\#}J$,
constructed by removing a short segment from each of $K$ and $J$ and joining each free end of $K$ to a different free end of $J$ to form a new knot. The connected sum of two knots always exists but is not necessarily unique.

The *connected sum of oriented knots* $K$ and $J$ is a connected sum of knots which has a consistent orientation^{} inherited from that of $K$ and $J$. This sum always exists and is unique.

###### Example.

Suppose $K$ and $J$ are both the trefoil knot.

By one choice of segment deletion and reattachment, $K\mathit{}\mathrm{\#}\mathit{}J$ is the quatrefoil knot.

Title | connected sum |
---|---|

Canonical name | ConnectedSum |

Date of creation | 2013-03-22 13:17:33 |

Last modified on | 2013-03-22 13:17:33 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 11 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 57M25 |

Synonym | knot sum |

Related topic | KnotTheory |