Lombardi drawings of knots and links

Philipp Kindermann, Stephen Kobourov, Maarten Löffler, Martin Nöllenburg, André Schulz, Birgit Vogtenhuber

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)444-476
Number of pages33
JournalJournal of Computational Geometry
Volume10
Issue number1
DOIs
StatePublished - Jan 1 2019

Fingerprint

Knot
Multigraph
Arc of a curve
Diagram
Projection
Angle
Simple Closed Curve
Graph Drawing
Curve
Drawing
Requirements
Arbitrary

ASJC Scopus subject areas

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

Cite this

Kindermann, P., Kobourov, S., Löffler, M., Nöllenburg, M., Schulz, A., & Vogtenhuber, B. (2019). Lombardi drawings of knots and links. 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.

In: Journal of Computational Geometry, Vol. 10, No. 1, 01.01.2019, p. 444-476.

Research output: Contribution to journalArticle

Kindermann, P, Kobourov, S, Löffler, M, Nöllenburg, M, Schulz, A & Vogtenhuber, B 2019, 'Lombardi drawings of knots and links', Journal of Computational Geometry, vol. 10, no. 1, pp. 444-476. https://doi.org/10.20382/jocg.v10i1a15
Kindermann P, Kobourov S, Löffler M, Nöllenburg M, Schulz A, Vogtenhuber B. Lombardi drawings of knots and links. Journal of Computational Geometry. 2019 Jan 1;10(1):444-476. https://doi.org/10.20382/jocg.v10i1a15
Kindermann, Philipp ; Kobourov, Stephen ; Löffler, Maarten ; Nöllenburg, Martin ; Schulz, André ; Vogtenhuber, Birgit. / Lombardi drawings of knots and links. In: Journal of Computational Geometry. 2019 ; Vol. 10, No. 1. pp. 444-476.
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