Lombardi drawings of knots and links

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

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

1 Citation (Scopus)

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
PublisherSpringer-Verlag
Pages113-126
Number of pages14
ISBN (Print)9783319739144
DOIs
StatePublished - Jan 1 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

Fingerprint

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

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Kindermann, P., Kobourov, S. G., Löffler, M., Nöllenburg, M., Schulz, A., & Vogtenhuber, B. (2018). Lombardi drawings of knots and links. In 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.

Graph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers. Springer-Verlag, 2018. p. 113-126 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10692 LNCS).

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

Kindermann, P, Kobourov, SG, Löffler, M, Nöllenburg, M, Schulz, A & Vogtenhuber, B 2018, Lombardi drawings of knots and links. in 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
Kindermann P, Kobourov SG, Löffler M, Nöllenburg M, Schulz A, Vogtenhuber B. Lombardi drawings of knots and links. In Graph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers. Springer-Verlag. 2018. p. 113-126. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-73915-1_10
Kindermann, Philipp ; Kobourov, Stephen G ; Löffler, Maarten ; Nöllenburg, Martin ; Schulz, André ; Vogtenhuber, Birgit. / Lombardi drawings of knots and links. Graph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers. Springer-Verlag, 2018. pp. 113-126 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
@inproceedings{05b3e7cc5a28423c8c72dc9f8ae22817,
title = "Lombardi drawings of knots and links",
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.",
author = "Philipp Kindermann and Kobourov, {Stephen G} and Maarten L{\"o}ffler and Martin N{\"o}llenburg and Andr{\'e} Schulz and Birgit Vogtenhuber",
year = "2018",
month = "1",
day = "1",
doi = "10.1007/978-3-319-73915-1_10",
language = "English (US)",
isbn = "9783319739144",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
publisher = "Springer-Verlag",
pages = "113--126",
booktitle = "Graph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers",

}

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 -