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Mathematics and Statistics
Completed Project

The Turaev Genus of Torus Knots with Small Braid Index

Kaitian Jin ’18, Eli Polston ’18, Yanjie Zheng ’18, and Adam Lowrance (Mathematics and Statistics>)

A knot invariant is a property defined for all knots such that the invariant will be the same for equivalent knots. In particular, we looked at an invariant known as the Turaev genus. Essentially, the Turaev genus of a torus knot is a measure of complexity on the torus associated with the knot. In this project, we wanted to know the Turaev genus of torus knots, knots that lie flat on a torus. A torus knot is characterized by a pair of coprime integers, p and q. The (p,q)-torus knot intersects a cross section of the torus p times and wraps around the core circle of the torus q times. The braid index of the (p,q)-torus knot is said to be p. Previous research discovered the Turaev genus of torus knots with braid index two and three. The Turaev genus of torus knots with braid index four or greater was previously unknown. To find the Turaev genus of torus knots with greater braid index, we looked at the diagrams of the knots. A diagram of a knot is a projection of a knot onto the plane. The Turaev genus of a diagram gives the upper bound of the Turaev genus of a knot. We calculated the width of another knot invariant called knot Floer homology to get the lower bound of the Turaev genus of the knots. We transformed the diagram through braid moves to lower the upper bound so it equaled the lower bound given by the width of knot Floer homology. By this technique, we found the Turaev genus of the family of torus knots with braid index 4 and subsets of torus knots with braid index 5 and 6. We also developed some new techniques that will hopefully help further research determine the Turaev genus of torus knots with greater braid index.