2017/09/30 by Benjamin Bode
Computer Science · Mathematics · #1-planar graph #Additive function #Advanced Combinatorial Mathematics #Book embedding #Chordal graph #Combinatorics #Computational Geometry and Mesh Generation #Crossing number (knot theory) #Discrete mathematics #Embedding #Geometric and Algebraic Topology #Graph #Knot (papermaking) #Mathematical analysis #Mathematics #Planar graph #math.GT #msc:57M25
paper · pdf · doi:10.1016/j.topol.2018.05.001
published in Topology and its Applications 243, 33-51 (Elsevier BV) · 20 pages, 11 figures, changes from version1: added Lemma 5.2 and corrected mistake in Proposition 5.3, improved quality of figures
openalex publication_date 2018/05/09 · arxiv created 2019/03/15 · arxiv updated 2019/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the minimal crossing number c(K1# K2) of composite knots K1# K2, where K1 and K2 are prime, by relating it to the minimal crossing number of spatial graphs, in particular the 2n-theta curve θ_K1,K2n that results from tying n of the edges of the planar embedding of the 2n-theta graph into K1 and the remaining n edges into K2. We prove that for large enough n we have c(θK1,K2n)=n(c(K1)+c(K2)). 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(K1# K2).