Crossing numbers of composite knots and spatial graphs

We study the minimal crossing number $c(K_1\mathrm{\#}{K}_2)$ of composite knots $K_1\mathrm{\#}{K}_2$, where $K_1$ and $K_2$ are prime, by relating it to the minimal crossing number of spatial graphs, in particular the $2n$-theta curve ${\theta }_{K_1,K_2}^n$ that results from tying $n$ of the edges of the planar embedding of the $2n$-theta graph into $K_1$ and the remaining $n$ edges into $K_2$. We prove that for large enough $n$ we have $c({\theta }_{K_1,K_2}^n)=n(c(K_1)+c(K_2))$. 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 $c(K_1\mathrm{\#}{K}_2)$.