Seifert Surface 3D Print

Seifert Surface on Trefoil

Today's adventure is printing a Seifert surface on a trefoil knot! I'm sure you all know what that is (\sarcasm) If you want to know more about this surface, there is an explanation at the end.

When I was a first year in college, I did an independent study on knot theory and really loved it. For that project I made a model of Seifert surfaces on both a trefoil and figure 8 knot out of a wire hanger and panty hose. I wish I still had a picture. I have always wanted to carve a larger, more sturdy version out of wood, as a tiered cupcake stand. I got one step closer to this goal today; I 3D printed the thing!

When I read this post by Laura Taalman and Jonathan Gerhard, I realized I too could 3D print these surfaces! Laura and Jonathan used a wonderful program called SeifertView to model the surface of their knot. I downloaded it and got it running on my mac through wine bottler.  This program is so much fun! Check out some screen shots:

Trefoil Surface in SeifertView

Close Up

Laura and Jonathan did get a special version that would export to stl (although I only ever got it to export to .obj) in order to 3d print the surface. Clearly I needed this special version of SeifertView as the original doesn't have many export options. So, without ever meeting either of them, I emailed Jonathan (thinking it was Laura, I guess I didn't read the post that closely) and asked for it. He responded within a few hours! 

I then followed the suggestions from Jonathan and brought the .obj file into blender, and thickened the mesh using the solidify tool (here is an awesome tutorial on the tool) and exported as .stl.

Blender

From there I imported into makerware and printed! (thingiverse)

In Makerbot Makerware

Printing with rafts and supports. 0.3mm, 10% infill

Final Product! Would print at 0.2mm next time

About the Surface:

A knot in mathematics is not so much like the knot you tie on your shoe laces, rather it resembles more the knots of celtic art work. In fancy terms, a knot is an embedding of a loop in 3 dimensional space. An "untied" knot is just a circle. The simplest knot you can make is the trefoil. Here is a drawing of one:

Trefoil Knot

A knot surface is a surface whose boundary, or edge, is a knot. So for the unknot (or circle) the knot surface is a disk. The interesting question here is, what would a surface look like if its boundary were a knot. One option for a surface whose boundary is the trefoil knot is a strip with three half twists (pictured below).

Surface on Trefoil

The problem with this surface is that it has only one side! If you were an ant, placed on this strip, you could walk in a straight line and eventually end up under where you started without ever crossing the edge. See Vi Hart's video for more explanation.

The question remains: does every knot bound a surface that has two sides? Herbert Seifert found a solution to this problem. He devised an algorithm that would define a two-sided (orientable) surface on any knot!

The culmination of this one quarter was proving that the Seifert surface on a trefoil knot is isomorphic to a punctured torus. It would be really cool if I could somehow print the steps in the dissection proof... A project for another day.

Punctured Torus