Crossing numbers of composite knots and spatial graphs

Abstract

We study the minimal crossing number $c(K_1\#K_2)$ of composite knots $K_1\#K_2$, 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 $c(K_1\#K_2)$.

20 pages, 11 figures, changes from version1: added Lemma 5.2 and corrected mistake in Proposition 5.3, improved quality of figures

References in corpus (3)

• The crossing number of composite knots
• Estimates for the minimal crossing number
• On the question of genericity of hyperbolic knots