Connected Sum - Connected Sum of Knots

Connected Sum of Knots

There is a closely related notion of the connected sum of two knots. In fact, if one regards a knot merely as a one-manifold, then the connected sum of two knots is just their connected sum as a one-dimensional manifold. However, the essential property of a knot is not its manifold structure (all knots are circles) but rather its embedding into the ambient space. So the connected sum of knots has a more elaborate definition that produces a well-defined embedding, as follows.

This procedure results in the projection of a new knot, a connected sum (or knot sum, or composition) of the original knots. For the connected sum of knots to be well defined, one has to consider oriented knots in 3-space. To define the connected sum for two oriented knots:

1. Consider a planar projection of each knot and suppose these projections are disjoint.
2. Find a rectangle in the plane where one pair of sides are arcs along each knot but is otherwise disjoint from the knots and so that the arcs of the knots on the sides of the rectangle are oriented around the boundary of the rectangle in the same direction.
3. Now join the two knots together by deleting these arcs from the knots and adding the arcs that form the other pair of sides of the rectangle.

The resulting connected sum knot inherits an orientation consistent with the orientations of the two original knots, and the oriented ambient isotopy class of the result is well-defined, depending only on the oriented ambient isotopy classes of the original two knots.

Under this operation, oriented knots in 3-space form a commutative monoid with unique prime factorization, which allows us to define what is meant by a prime knot. Proof of commutativity can be seen by letting one summand shrink until it is very small and then pulling it along the other knot. The unknot is the unit. The two trefoil knots are the simplest prime knots. Higher dimensional knots can be added by splicing the -spheres.

In three dimensions, the unknot cannot be written as the sum of two non-trivial knots. This fact follows from additivity of knot genus; another proof relies on an infinite construction sometimes called the Mazur swindle. In higher dimensions (with codimension at least three), it is possible to get an unknot by adding two nontrivial knots.

If one does not take into account the orientations of the knots, the connected sum operation is not well defined on isotopy classes of (nonoriented) knots. To see this, consider two noninvertible knots K, L which are not equivalent (as unoriented knots); for example take the two pretzel knots K = P(3,5,7) and L = P(3,5,9). Let K₊ and K_- be K with its two inequivalent orientations, and let L₊ and L_- be L with its two inequivalent orientations. There are four oriented connected sums we may form:

• A = K₊ # L₊
• B = K_- # L_-
• C = K₊ # L_-
• D = K_- # L₊

The oriented ambient isotopy classes of these four oriented knots are all distinct. And, when one considers ambient isotopy of the knots without regard to orientation, there are two distinct equivalence classes: { A ~ B } and { C ~ D }. To see that A and B are unoriented equivalent, simply note that they both may be constructed from the same pair of disjoint knot projections as above, the only difference being the orientations of the knots. Similarly, one sees that C and D may be constructed from the same pair of disjoint knot projections.