%0 Journal Article %1 kklnsv-ldkl-jocg19 %A Kindermann, Philipp %A Kobourov, Stephen %A Löffler, Maarten %A Nöllenburg, Martin %A Schulz, André %A Vogtenhuber, Birgit %D 2019 %J Journal of Computational Geometry %K imported myown %N 1 %P 444--476 %R 10.20382/jocg.v10i1a15 %T Lombardi Drawings of Knots and Links %V 10 %X 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 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.

@article{kklnsv-ldkl-jocg19, 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.}, added-at = {2019-10-01T18:14:41.000+0200}, arxiv = {https://arxiv.org/abs/1708.09819}, author = {Kindermann, Philipp and Kobourov, Stephen and L{\"o}ffler, Maarten and N{\"o}llenburg, Martin and Schulz, Andr{\'e} and Vogtenhuber, Birgit}, biburl = {https://www.bibsonomy.org/bibtex/2835fffd0fc99dc9571bdbb6936c8ad98/kindermann}, doi = {10.20382/jocg.v10i1a15}, interhash = {0d00ccfc0feda1779f71eb3eb9704465}, intrahash = {835fffd0fc99dc9571bdbb6936c8ad98}, journal = {Journal of Computational Geometry}, keywords = {imported myown}, number = 1, pages = {444--476}, slides = {http://www1.pub.informatik.uni-wuerzburg.de/pub/kindermann/slides/2017-gd-knots-maarten.pdf}, timestamp = {2021-02-12T10:11:04.000+0100}, title = {Lombardi Drawings of Knots and Links}, volume = 10, year = 2019 }