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Titel:

Lombardi Drawings of Knots and Links

AutorInnen: Philipp Kindermann
Stephen Kobourov
Maarten Löffler
Martin Nöllenburg
André Schulz
Birgit Vogtenhuber
Kategorie: Konferenzbandbeiträge
erschienen in: Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD'17). To appear.
Abstract:

Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into R2, such that no more than two points project to the same point in R2. These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in R3, so their projections should be smooth curves in R2 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 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.

BibTeX-Eintrag: @InProceedings{kklnsv-ldkl-gd17, author = {Philipp Kindermann and Stephen Kobourov and Maarten L{\"o}ffler and Martin N{\"o}llenburg and Andr{\'e} Schulz and Birgit Vogtenhuber}, title = {Lombardi Drawings of Knots and Links}, booktitle = {Proc. 25th International Symposium on Graph Drawing and Network Visualization (GD'17)}, year = {2017}, editor = {Fabrizio Frati and Kwan-Liu Ma}, note = {To appear.}, abstract = {Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into $R^2$, such that no more than two points project to the same point in $\R^2$. These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in $R^3$, so their projections should be smooth curves in $R^2$ 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 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 \emph{near-Lombardi}, that is, it can be drawn as Lombardi drawing when relaxing the angular resolution requirement by an arbitrary small angular offset $\eps$, while maintaining a $180^\circ$ angle between opposite edges.}, }
Philipp Kindermann | 04.09.2017
FernUni-Logo FernUniversität in Hagen, LG Theoretische Informatik, 58084 Hagen