Lombardi drawings of knots and links

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

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

Abstract

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.

Original languageEnglish (US)
Title of host publicationGraph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers
EditorsKwan-Liu Ma, Fabrizio Frati
PublisherSpringer-Verlag
Pages113-126
Number of pages14
ISBN (Print)9783319739144
DOIs
StatePublished - 2018
Event25th International Symposium on Graph Drawing and Network Visualization, GD 2017 - Boston, United States
Duration: Sep 25 2017Sep 27 2017

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10692 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other25th International Symposium on Graph Drawing and Network Visualization, GD 2017
CountryUnited States
CityBoston
Period9/25/179/27/17

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Fingerprint Dive into the research topics of 'Lombardi drawings of knots and links'. Together they form a unique fingerprint.

  • Cite this

    Kindermann, P., Kobourov, S., Löffler, M., Nöllenburg, M., Schulz, A., & Vogtenhuber, B. (2018). Lombardi drawings of knots and links. In K-L. Ma, & F. Frati (Eds.), 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