[Topology and Geometry Seminar] “Coiled Surfaces And The Slope Conjectures” by Dr Josh Howie (Monash University)
The seminar aims to introduce research topics in topology, geometry and its interactions with other sciences. Anyone interested in mathematics is welcome to attend.
This week, Dr Josh Howie (Monash University) will discuss "Coiled Surfaces And The Slope Conjectures".
Abstract: There are several conjectural relationships between the coloured Jones polynomial which is a quantum knot invariant, and the 3-dimensional geometry and topology of knot complements. The slope conjectures of Garoufalidis and Kalfagianni-Tran predict that the highest and lowest degrees of the coloured Jones polynomial contain topological information about essential surfaces properly embedded in the knot exterior. We introduce the class of coiled knots and use the closed fake surfaces of Ozawa to prove that their coiled surfaces are essential. This allows us to complete the proof that all knots up to 9 crossings satisfy the strong slope conjecture.