Bridge position and the representativity of spatial graphs

Makoto Ozawa

Published 2009-09-07, updated 2010-11-17Version 3

First, we extend Otal's result for the trivial knot to trivial spatial graphs, namely, we show that for any bridge tangle decomposing sphere $S^2$ for a trivial spatial graph $\Gamma$, there exists a 2-sphere $F$ such that $F$ contains $\Gamma$ and $F$ intersects $S^2$ in a single loop. Next, we introduce two invariants for spatial graphs. As a generalization of the bridge number for knots, we define the {\em bridge string number} $bs(\Gamma)$ of a spatial graph $\Gamma$ as the minimal number of $|\Gamma\cap S^2|$ for all bridge tangle decomposing sphere $S^2$. As a spatial version of the representativity for a graph embedded in a surface, we define the {\em representativity} of a non-trivial spatial graph $\Gamma$ as \[ r(\Gamma)=\max_{F\in\mathcal{F}} \min_{D\in\mathcal{D}_F} |\partial D\cap \Gamma|, \] where $\mathcal{F}$ is the set of all closed surfaces containing $\Gamma$ and $\mathcal{D}_F$ is the set of all compressing disks for $F$ in $S^3$. Then we show that for a non-trivial spatial graph $\Gamma$, \[ \displaystyle r(\Gamma)\le \frac{bs(\Gamma)}{2}. \] In particular, if $\Gamma$ is a knot, then $r(\Gamma)\le b(\Gamma)$, where $b(\Gamma)$ denotes the bridge number. This generalizes Schubert's result on torus knots.