## Abstract

We study the minimal crossing number c(K_{1}#K_{2}) of composite knots K_{1}#K_{2}, where K_{1} and K_{2} are prime, by relating it to the minimal crossing number of spatial graphs, in particular the 2n-theta-curve θ_{K1,K2 } ^{n} that results from tying n of the edges of the planar embedding of the 2n-theta-graph into K_{1} and the remaining n edges into K_{2}. We prove that for large enough n we have c(θ_{K1,K2 } ^{n})=n(c(K_{1})+c(K_{2})). 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(K_{1}#K_{2}).

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