As with ordinary primes, there is a unique way to construct knots from prime knots, but only if one specifies the order of the decomposition. In other words, knots are a kind of non-commutative prime.
The product operation for a torus knot, for example, involves a connecting cylinder surface containing two strands, which replace a small segment from each of the two knots. At the boundaries of this cylinder, the strands form two marked points on the loop of the knot, which are the ends of the cut out segment.
As Bar-Natan showed, one can also consider non-associative tangles. This paper contained the first attempt at defining the polytopes of Batanin’s higher operads.
Consider this lovely image of the Seifert surface of a trefoil, created using SeifertView by Jarke J. van Wijk.
Arcadian Functor 
Mahndisa S. Rigmaiden said,
March 12, 2007 @ 4:37 pm
03 12 07
Thanks Kea.
Kea said,
March 12, 2007 @ 6:46 pm
Oh, are you better, Mahndisa? Or getting there?
Doug said,
March 13, 2007 @ 2:32 am
Thanks for the fascinating SeifertView by Jarke J van Wijk.
I just realized that my architectural geometry perspective might be consistent with U-duality, as I understand it. Maybe Kneemo would find this of interest considering the name of his blog.