M Theory Lesson 24
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.