Arc index and Turaev genus

Adam Lowrance (Vassar)

Abstract: An arc presentation of a knot is an embedding of the knot in finitely many pages of an open-book decomposition such that the portion of the knot on each page is a single simple arc. The arc index of a knot is the minimum number of pages in any arc presentation of the knot. The Turaev surface of a knot diagram D is constructed by first taking the cobordism between the all-A and all-B states of D with saddle points corresponding to the crossings of D, and then capping off the boundary components of that cobordism with disks. The Turaev genus of a knot is the minimum genus of the Turaev surface for any diagram. In this talk, we discuss a conjecture relating the crossing number, arc index, and Turaev genus of a knot. This is joint work with Alvaro Del Valle Vilchez.