Crossing numbers of composite knots and spatial graphs

We study the minimal crossing number c(K1#K2) of composite knots K1#K2, where K1 and K2 are prime, by relating it to the minimal crossing number of spatial graphs, in particular the 2n-theta-curve θK1,K2n that results from tying n of the edges of the planar embedding of the 2n-theta-graph into K1 and the remaining n edges into K2. We prove that for large enough n we have c(θK1,K2n)=n(c(K1)+c(K2)). We also formulate additional relations between the crossing numbers of certain spatial graphs that, if satisfied, imply the additivity of the crossing number or at least give a lower bound for c(K1#K2).