### 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) |
---|---|

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 |

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### ASJC Scopus subject areas

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

### Cite this

*Journal of Computational Geometry*,

*10*(1), 444-476. https://doi.org/10.20382/jocg.v10i1a15

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

Research output: Contribution to journal › Article

*Journal of Computational Geometry*, vol. 10, no. 1, pp. 444-476. https://doi.org/10.20382/jocg.v10i1a15

}

TY - JOUR

T1 - Lombardi drawings of knots and links

AU - Kindermann, Philipp

AU - Kobourov, Stephen

AU - Löffler, Maarten

AU - Nöllenburg, Martin

AU - Schulz, André

AU - Vogtenhuber, Birgit

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into lR2, such that no more than two points project to the same point in lR2. These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in lR3, so their projections should be smooth curves in lR2 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.

AB - Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into lR2, such that no more than two points project to the same point in lR2. These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in lR3, so their projections should be smooth curves in lR2 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.

UR - http://www.scopus.com/inward/record.url?scp=85075318460&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075318460&partnerID=8YFLogxK

U2 - 10.20382/jocg.v10i1a15

DO - 10.20382/jocg.v10i1a15

M3 - Article

AN - SCOPUS:85075318460

VL - 10

SP - 444

EP - 476

JO - Journal of Computational Geometry

JF - Journal of Computational Geometry

SN - 1920-180X

IS - 1

ER -