Bridge position and the representativity of spatial graphs (Q409544)

A spatial graph is an embedding of an abstract graph into \(S^3\). Given a closed surface \(F\) in \(S^3\) and a non-planar spatial graph \(\Gamma\) on \(F\), \(r(F,\Gamma)\) is the minimal number of intersection points between \(\Gamma\) and the boundary of a compressing disc for \(F\). In this paper, the representativity \(r(\Gamma)\) of \(\Gamma\) is defined to be the maximum value of \(r(F,\Gamma)\) over all closed surfaces containing \(\Gamma\). The paper gives results relating the representativity to other properties and invariants of a spatial graph. In particular, it is shown that \(2r(\Gamma)\) is at most the bridge string number \(bs(\Gamma)\). This is the minimum intersection number of \(\Gamma\) with a sphere dividing it into two `trivial' tangles.NEWLINENEWLINEThe proofs in this paper are mostly direct. Some work by putting a spatial graph and a surface containing it into a suitable Morse position. Results of \textit{R. H. Fox} [Ann. Math. (2) 49, 462--470 (1948; Zbl 0032.12502)], \textit{M. Scharlemann} and \textit{A. Thompson} [Proc. Am. Math. Soc. 133, No. 6, 1573--1580 (2005; Zbl 1071.57015)], and \textit{F. Bonahon} [Ann. Sci. Éc. Norm. Supér. (4) 16, 237--270 (1983; Zbl 0535.57016)] regarding Heegaard splittings are also used.