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Wednesday, May 21, 2014

Day 268 - Tritangentless conformation of 3_1

The trefoil knot 3_1 is the only three-crossing knot, which means that every other knot that can be drawn as a projection in the plane with three crossings (and no fewer) must be topologically equivalent to a trefoil knot. In my opinion the coolest thing about the trefoil is that it has a tritangentless conformation. In such a conformation, there are no planes that are tangent to three points on the knot simultaneously. For those planes on the exterior of the knot this means that if you put the knot on a table, the knot will only touch the table in at most two places. On the left in the picture below is shown a standard trefoil conformation, and on the right is the same trefoil knot but in a tritangentless conformation (the same "rocking knot" model that we printed on Day 110 and Day 151):

Thingiverse link: http://www.thingiverse.com/make:78751

Settings: Replicator 2 with custom knot slicing/support profile.

Technical notes, see previous posts: See Day 110 for the parametric equations that describe this tritangentless trefoil conformation, a link to the paper those equations were found in, and details about the custom slicing profile. See Day 151 for the OpenSCAD code that will generate this knot without the "seam" that appears if you try to model it in Mathematica. For other interesting conformations of the trefoil see the Trefoil Menagerie on Thingiverse.