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.

Figure 1: The trefoil knot

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

Figure 2: $K\mathit{}\mathrm{\#}\mathit{}J$ 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