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Lombardi Drawings of Knots and Links

  • Philipp Kindermann
  • Stephen Kobourov
  • Maarten Löffler
  • Martin Nöllenburg
  • André Schulz
  • Birgit Vogtenhuber
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10692)

Abstract

Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into \({\mathsf{l}\mathsf{R}}^2\), such that no more than two points project to the same point in \({\mathsf{l}\mathsf{R}}^2\). These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in \({\mathsf{l}\mathsf{R}}^3\), so their projections should be smooth curves in \({\mathsf{l}\mathsf{R}}^2\) 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 \(\varepsilon \), while maintaining a \(180^\circ \) angle between opposite edges.

Notes

Acknowledgements

Research for this work was initiated at Dagstuhl Seminar 17072 Applications of Topology to the Analysis of 1-Dimensional Objects which took place in February 2017. We thank Benjamin Burton for bringing the problem to our attention and Dylan Thurston for helpful discussion.

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Philipp Kindermann
    • 1
  • Stephen Kobourov
    • 2
  • Maarten Löffler
    • 3
  • Martin Nöllenburg
    • 4
  • André Schulz
    • 1
  • Birgit Vogtenhuber
    • 5
  1. 1.FernUniversität in HagenHagenGermany
  2. 2.University of ArizonaTucsonUSA
  3. 3.Universiteit UtrechtUtrechtThe Netherlands
  4. 4.TU WienViennaAustria
  5. 5.Graz University of TechnologyGrazAustria

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