### Abstract

Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into IR^{2} such that no more than two points project to the same point in IR^{2} These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in IR^{2} so their projections should be smooth curves in IR^{2} with good continuity and large crossing angles: exactly the properties of Lombardi graph drawings (defined by circular-arc edges and perfect angular resolution). We show that several knots do not allow plane Lombardi drawings. On the other hand, we identify a large class of 4-regular plane multigraphs that do have Lombardi drawings. We then study two relaxations of Lombardi drawings and show that every knot admits a 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 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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Title of host publication | Graph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers |

Publisher | Springer-Verlag |

Pages | 113-126 |

Number of pages | 14 |

ISBN (Print) | 9783319739144 |

DOIs | |

State | Published - Jan 1 2018 |

Event | 25th International Symposium on Graph Drawing and Network Visualization, GD 2017 - Boston, United States Duration: Sep 25 2017 → Sep 27 2017 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 10692 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 25th International Symposium on Graph Drawing and Network Visualization, GD 2017 |
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Country | United States |

City | Boston |

Period | 9/25/17 → 9/27/17 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Graph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers*(pp. 113-126). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10692 LNCS). Springer-Verlag. https://doi.org/10.1007/978-3-319-73915-1_10

**Lombardi drawings of knots and links.** / Kindermann, Philipp; Kobourov, Stephen G; Löffler, Maarten; Nöllenburg, Martin; Schulz, André; Vogtenhuber, Birgit.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Graph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10692 LNCS, Springer-Verlag, pp. 113-126, 25th International Symposium on Graph Drawing and Network Visualization, GD 2017, Boston, United States, 9/25/17. https://doi.org/10.1007/978-3-319-73915-1_10

}

TY - GEN

T1 - Lombardi drawings of knots and links

AU - Kindermann, Philipp

AU - Kobourov, Stephen G

AU - Löffler, Maarten

AU - Nöllenburg, Martin

AU - Schulz, André

AU - Vogtenhuber, Birgit

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into IR2 such that no more than two points project to the same point in IR2 These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in IR2 so their projections should be smooth curves in IR2 with good continuity and large crossing angles: exactly the properties of Lombardi graph drawings (defined by circular-arc edges and perfect angular resolution). We show that several knots do not allow plane Lombardi drawings. On the other hand, we identify a large class of 4-regular plane multigraphs that do have Lombardi drawings. We then study two relaxations of Lombardi drawings and show that every knot admits a 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 Lombardi drawing when relaxing the angular resolution requirement by an arbitrary small angular offset ε while maintaining a 180° angle between opposite edges.

AB - Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into IR2 such that no more than two points project to the same point in IR2 These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in IR2 so their projections should be smooth curves in IR2 with good continuity and large crossing angles: exactly the properties of Lombardi graph drawings (defined by circular-arc edges and perfect angular resolution). We show that several knots do not allow plane Lombardi drawings. On the other hand, we identify a large class of 4-regular plane multigraphs that do have Lombardi drawings. We then study two relaxations of Lombardi drawings and show that every knot admits a 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 Lombardi drawing when relaxing the angular resolution requirement by an arbitrary small angular offset ε while maintaining a 180° angle between opposite edges.

UR - http://www.scopus.com/inward/record.url?scp=85041821210&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85041821210&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-73915-1_10

DO - 10.1007/978-3-319-73915-1_10

M3 - Conference contribution

SN - 9783319739144

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 113

EP - 126

BT - Graph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers

PB - Springer-Verlag

ER -