# connected sum

The connected sum of knots $K$ and $J$ is a knot, denoted by $K\#J$, 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, $K\#J$ is the quatrefoil knot.

Title connected sum ConnectedSum 2013-03-22 13:17:33 2013-03-22 13:17:33 Mathprof (13753) Mathprof (13753) 11 Mathprof (13753) Definition msc 57M25 knot sum KnotTheory