%0 Conference Paper %1 kklnsv-ldkl-gd17 %A Kindermann, Philipp %A Kobourov, Stephen %A Löffler, Maarten %A Nöllenburg, Martin %A Schulz, André %A Vogtenhuber, Birgit %B Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD'17) %D 2017 %E Frati, Fabrizio %E Ma, Kwan-Liu %I Springer %K myown %P 113--126 %R 10.1007/978-3-319-73915-1_10 %T Lombardi Drawings of Knots and Links %V 10692 %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 $\epsilon$, while maintaining a $180^\circ$ angle between opposite edges.

@inproceedings{kklnsv-ldkl-gd17, 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 near-Lombardi, that is, it can be drawn as Lombardi drawing when relaxing the angular resolution requirement by an arbitrary small angular offset $\epsilon$, while maintaining a $180^\circ$ angle between opposite edges.}, added-at = {2017-07-30T22:27:36.000+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/26065a1bf3d12df378b7c94ac4c2cb952/kindermann}, booktitle = {Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD'17)}, doi = {10.1007/978-3-319-73915-1_10}, editor = {Frati, Fabrizio and Ma, Kwan-Liu}, interhash = {e12161e0c0de18241afcf2a9e269dcdf}, intrahash = {6065a1bf3d12df378b7c94ac4c2cb952}, keywords = {myown}, month = sep, pages = {113--126}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, slides = {http://www1.pub.informatik.uni-wuerzburg.de/pub/kindermann/slides/2017-gd-knots-maarten.pdf}, timestamp = {2018-09-18T07:10:02.000+0200}, title = {Lombardi Drawings of Knots and Links}, volume = 10692, year = 2017 }