Crossing numbers of composite knots and spatial graphs (Q1637160)

This paper relates the minimal crossing number of composite prime knots to that of certain spatial graphs, pointing out that it is not even known that adding a knot cannot reduce it, and that if it is always additive (proven, thus far, for torus and adequate knots) that would disprove the more recent conjecture that the percentage of hyperbolic knots among all prime knots of minimal crossing number at most \(n\) approaches 100 as \(n\) goes to infinity. (Fans of hyperbolic knot theory, take notice.) Looking at the curve that results from tying \(n\) of the edges of the planar embedding of a \(2n\)-theta-graph into one component of a product knot and the remaining \(n\) edges into the other, the author proves that for large enough \(n\), the crossing number of such a curve is \(n\) times the sum of the crossing numbers of the knots. Additional relations are formulated between crossing numbers of other graphs that, if satisfied, imply crossing number additivity, ``or at least give a lower bound'' for it.