Morse Structures on Rational Open Books

Abstract: A knot in 3-dimensional Euclidean space can be projected to a 2-dimensional plane to obtain a knot diagram, which recovers the original knot if we record crossing information. In contact geometry, one studies knots that are constrained to be tangent to a specified plane field. It turns out that for such knots, the projection to the yz-plane loses no information: the missing coordinate can be recovered exactly from the slope of the projected curve, and crossing data need not be specified. However, it is not clear how to extend this idea to arbitrary 3-manifolds: what surfaces do we project to, and what would "projection" even mean? In this talk, I will explain how open book decompositions lead to suitable embedded tori, and how Morse-theoretic data can be used to project to these tori, following work of David Gay and Joan Licata. I will then describe how to extend this picture to rational open books, and we will see how the rational viewpoint simplifies the story for lens spaces.