Crossing numbers of composite knots and spatial graphs

Benjamin Bode

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Abstract

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 θK1,K2 n 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,K2 n)=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).

Original languageEnglish
Pages (from-to)33-51
Number of pages19
JournalTopology and its Applications
Volume243
Early online date9 May 2018
DOIs
Publication statusPublished - 1 Jul 2018

Structured keywords

  • SPOCK

Keywords

  • Composite knots
  • Crossing number
  • Spatial graphs
  • Theta-curves

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