## Abstract

Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into lR^{2}, such that no more than two points project to the same point in lR^{2}. These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in lR^{3}, so their projections should be smooth curves in lR^{2} with good continuity and large crossing angles: exactly the properties of Lombardi graph drawings (defned by circular-arc edges and perfect angular resolution). We show that several knots do not allow crossing-minimal plane Lombardi drawings. On the other hand, we identify a large class of 4-regular plane multigraphs that do have plane Lombardi drawings. We then study two relaxations of Lombardi drawings and show that every knot admits a crossing-minimal plane 2-Lombardi drawing (where edges are composed of two circular arcs). Further, every knot is near-Lombardi, that is, it can be drawn as a plane Lombardi drawing when relaxing the angular resolution requirement by an arbitrary small angular offset ε, while maintaining a 180^{◦} angle between opposite edges.

Original language | English (US) |
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Pages (from-to) | 444-476 |

Number of pages | 33 |

Journal | Journal of Computational Geometry |

Volume | 10 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2019 |

## ASJC Scopus subject areas

- Geometry and Topology
- Computer Science Applications
- Computational Theory and Mathematics