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Owner confidence rating: Very high Entry average rating: No information on entry rating
connected sum (Definition)

The connected sum of knots $ K$ and $ J$ is a knot, denoted by $ K\char93 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: The trefoil knot
\includegraphics{TrefoilKnotPF2007}
By one choice of segment deletion and reattachment, $ K\char93 J$ is the quatrefoil knot.
Figure: $ K\char93 J$ is the quatrefoil knot
\includegraphics{QuatrefoilKnotPF2007}



"connected sum" is owned by Mathprof. [ full author list (3) | owner history (4) ]
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See Also: knot theory

Other names:  knot sum
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Cross-references: sum, orientation, consistent, oriented, segment, knots
There are 6 references to this entry.

This is version 8 of connected sum, born on 2002-12-19, modified 2007-05-26.
Object id is 3780, canonical name is ConnectedSum.
Accessed 4414 times total.

Classification:
AMS MSC: 57M25 (Manifolds and cell complexes :: Low-dimensional topology :: Knots and links in $S^3$)
Pending Errata and Addenda
None.
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Connected sum graphics by PrimeFan on 2007-05-26 15:05:49
After mps's correction to make the knot graphics "prettier" (meaning smoother with no unnecessary cusps) I thought I could just redraw them by hand. Unfortunately, my hands just aren't as steady as they used to be. In exchange for no arthritis pain I pay the price in shakiness and unsteadiness. I hope a young man with drawing skills will pick up this entry and make much better looking graphics. I don't mind if you delete my graphics, though you may want to keep them for reference until you get the better graphics in place.
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