Cross sections of star-crossed knots

Date started: November 2019
Leads: Laura Taalman

Abstract

We used OpenSCAD sweeper code to produce a model of the knot 8_19 with a star-shaped cross section, then extracted SVG images of planar cross sections from both symmetric and non-symmetric knot orientations in R3. These cross sections are interesting from a mathematical perspective, as they aid in understanding the internal structure of the knot. In addition we can use these cross sections to create stylized motifs to use in surface design patterns. Future directions include exploration of knot complements using laser cut slices.

Media

 

Early stage of 3D print showing the star cross-sections
Main supports removed; preparing to dissolve interface support
Finished model of star-crossed knot 8_18
Slices of star-crossed knot 8_18 from a symmetric orientation
Slices of star-crossed knot 8_18 from a non-symmetric orientation
Selected 3D elements
Motifs taken from selected 3D elements of knot slices
Half-drop surface design pattern with two repeat scales

Next steps are to explore sliced star-crossed knot complements using a laser cutter and transparent acrylic, in the spirit of this simpler trefoil example from our sliced knot complement project:

Laser cut slices of the complement of a tritangentless trefoil knot