connected sum

The connected sumPlanetmathPlanetmath 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 orientationPlanetmathPlanetmath inherited from that of K and J. This sum always exists and is unique.


Suppose K and J are both the trefoil knot.

Figure 1: The trefoil knot

By one choice of segment deletion and reattachment, K#J is the quatrefoil knot.

Figure 2: K#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