Adams
does a nice job of explaining how knots can be composed. He describes
composition of knots like this: a new knot can be made from two other knots,
called knot J and knot K, by taking each knot projection, removing a small
arc from each, and connecting the four endpoints, as shown in figure 22.
The two arcs that are chosen are assumed to not overlap and not have crossings
between themselves or the original knot projections.[1]
See figure 23 for an example of how NOT to
compose two knots. The result if done correctly is called the composition
of the two knots. Adams denotes the composition as J#K. [1]
Just like in the integers, a knot K is called
composite if two nontrivial knots J1 and J2 (that is, knots that are not
the unknot) exist which, when composed, produce K. J1 and J2 are
called the factor knots. As you can easily guess, a prime knot only
has as factors itself and the unknot. See figure
24 for some examples of prime knots.
A
rather interesting point Adams makes is that if the unknot happens to be
composite, then every knot must be composite. It may seem obvious
at first thought that the unknot is not composite, but figure 25 shows
an unknot that looks as if it could be composite. If the unknot is
composite, then it must be made up of two nontrivial knots. That
is, every knot which is the composition of itself with the unknot would
really be a composition of two nontrivial knots, and hence be composite.
So every knot would be composite. However, Adams proves that this
is not true, just as the analogous situation in the integers is not true,
where 1 is not a composite number. [1]
I make one last point by noting that this analogy between composites and primes in the integers and in knots continues by noting the theorem stating the uniqueness and existence of a decomposition of knots by Murasugi:
(1) Any knot can be decomposed into a finite number of prime knots.A proof of this is very involved and can be found in Burde and Kieschang's book. [2]
(2) This decomposition, excluding the order, is unique. That is, suppose we can decompose K in two ways: K1, K2, ..., Km, and K1`, K2`, ..., Kn`. Then n = m, and, furthermore, if we suitably choose the numbering of K1, K2, ..., Km, then K1 = K1`, ..., Kn = Kn`. [4]
