Computations in Knot Theory*
Adam Lowrance (Mathematics and Statistics)
In knot theory, we study mathematical models of knotted ropes in space. The knotted rope forms a closed path (so it has no loose ends), it is infinitely thin, and it is infinitely stretchable. Two knots are considered equivalent if the first can be deformed into the second by stretching, bending, and moving our idealized rope through space. The rope cannot be cut and glued back together during the deformation and one strand is not allowed to pass through another.
Shining a flashlight at a knot leaves a shadow of the knot on the wall. If the flashlight is held at just the right position, then at most two strands cross at one time. When drawing the shadow of the knot on paper, we indicate which of the two strands at a crossing is closer to the flashlight. Such a drawing of a knot is called a knot diagram. Every knot has infinitely many diagrams.
In this project, we will study certain quantities associated to knot diagrams. Our goal will be to minimize these quantities over all possible diagrams of the knot. We use techniques are pictorial, combinatorial, and algebraic in nature.
Prerequisites: Students applying to this project should have some experience with writing proofs. Courses that would satisfy this requirement include Math 261, 263, 321, 331, 324, 361, and 364. If a student has proof-writing experience, but does not have one of the above courses, the student should feel free to apply to this project.
How should students express their interest in this project? Interested students should email Professor Lowrance to set up an interview.