2009/09/07 by Makoto Ozawa, Ozawa, Makoto
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Primary 57M25 #Secondary 57Q35 #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.0909.1162
openalex publication_date 2009/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
First, we extend Otal's result for the trivial knot to trivial spatial graphs, namely, we show that for any bridge tangle decomposing sphere S2 for a trivial spatial graph Γ, there exists a 2-sphere F such that F contains Γ and F intersects S2 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(Γ) of a spatial graph Γ as the minimal number of |Γ∩ S2| for all bridge tangle decomposing sphere S2. 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 Γ as r(Γ)=maxF\inF minD\inDF |∂ D∩ Γ|, where F is the set of all closed surfaces containing Γ and DF is the set of all compressing disks for F in S3. Then we show that for a non-trivial spatial graph Γ, r(Γ)≤ (bs(Γ))/(2). In particular, if Γ is a knot, then r(Γ)≤ b(Γ), where b(Γ) denotes the bridge number. This generalizes Schubert's result on torus knots.