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 language | English |
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Pages (from-to) | 33-51 |
Number of pages | 19 |
Journal | Topology and its Applications |
Volume | 243 |
Early online date | 9 May 2018 |
DOIs | |
Publication status | Published - 1 Jul 2018 |
Structured keywords
- SPOCK
Keywords
- Composite knots
- Crossing number
- Spatial graphs
- Theta-curves