Understanding the Torus and Trefoil Knot

Torus and Trefoil Knot

This preliminary visualization demonstrates how a square can be glued along its oposite edges to produce a torus.

Use the sliders in the parameters panel to manipulate the visualization. Gluing along either the latitudinal or longitudinal will transform the square into a vertical or horizontal cylinder. Finally gluing the remaining circuluar edges of the cylinder produces a torus.

As defined previously, we consider a knot embedded in the 3-sphere, \(K \in S^3 \). A special kind of knots which are always fibred are called torus knots, which are embedded in the surface of a torus. In the following visualizations we consider the trefoil knot or the (2,3)-torus knot. Note the (2,3)-torus knot is named because it touches the latitudinal edge 2 times and the longitudinal edge 3 times. Finally, trefoil traveller demonstrates how the knot connects along the edges.

About Project

This is my final deliverable for the summer research component of a Master's in Mathematics at the University of Toronto supervised by Dr. Dror Bar-Natan.

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About Me

My name is Jesse Bettencourt and I've recently acquired an M.Sc. in Mathematics from U of T. I'm interested in vizualizations from the intersection of math and computer science.