Speaker: Christine Lee, University of Texas, Austin

Title: Jones slopes and Murasugi sums of links

Date: 02/09/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

A Jones surface for a knot in the three-sphere is an essential surface whose boundary slopes, Euler characteristic, and number of sheets correspond to quantities defined from the asymptotics of the degrees of colored Jones polynomial. The Strong Slope Conjecture by Garoufalidis and Kalfagianni-Tran predicts that there are Jones surfaces for every knot.
A link diagram D is said to be a Murasugi sum of two links D' and D' if a state graph of D has a cut vertex, which separates the graph into two state graphs of D' and D', respectively. We may obtain a state surface in the complement of the link K represented by D by gluing the state surface for D and the state surface for D' along the disk filling the circle represented by the cut vertex in the state graph. The resulting surface is called the Murasugi sum of the two state surfaces.
We consider near-adequate links which are Murasugi sums of certain non-adequate link diagrams with an adequate link diagram along their all-A state graphs with an additional graphical constraint. For a near-adequate knot, the Murasugi sum of the corresponding state surface is a Jones surface by the work of Ozawa. We discuss how this proves the Strong Slope Conjecture for this class of knots and raises interesting questions about constraints on the possible Murasugi sum-decompositions of a link diagram.