Joseph Geisz: Stick Knots

Stick Knots

My current research involves knot theory - the study of loops in 3D space. In particular I am interested in stick knots (also called piecewise linear knots) and random knot models, methods of randomly sampling knots.

It is known that the only knot types that can be constructed with 6 segments are the unknot and trefoil knot. Similar lists are known for 7 and 8. In my thesis I will answer the following question: What knot types can be made with only 9 linear segments? In this research I use linear programming to find realizations of 2D polygonal knot diagrams. See this visualization to understand more of this process

One question we have answered is the following: choose 6 points uniformly at random from the sphere. Connect them cyclically with linear edges. In our preprint we show that the probability that such a knot is trefoil is exactly 1/(16π^2). You can play with knotted hexagons here

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