# 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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## 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