2008/08/31 by Youngsik Huh, Ryo Nikkuni
Computer Science · Mathematics · #1-planar graph #Advanced Graph Theory Research #Artificial intelligence #Book embedding #Combinatorics #Computational Geometry and Mesh Generation #Computer science #Discrete mathematics #Embedding #Geometric and Algebraic Topology #Graph #Graph embedding #Immersion (mathematics) #Line graph #Mathematics #Outerplanar graph #Planar graph #Planar straight-line graph #Pure mathematics #Voltage graph #math.GT #msc:57M15 #msc:57M25
paper · pdf · doi:10.1142/s0218216510008261
published as J. Knot Theory Ramifications 19 (2010), 917--933 · 16 pages, 31 figures
arxiv created 2009/05/07 · openalex publication_date 2010/07/01 · arxiv updated 2020/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper, we show that if a generic immersion of a planar graph is knotted then the number of double points of the immersion is more than or equal to three. To prove this, we also show that an embedding of a graph obtained from a generic immersion of the graph (does not need to be planar) with at most three double points is totally free if it contains neither a Hopf link nor a trefoil knot.