| Name: | Zheheng (Tony) Xiao |
|---|---|
| Email: | zx2377 (at) columbia.edu |
| Office: | CORE 444 |
| Home Institution: | Columbia University |
| Project: | Concordance Invariants of Satellite Knots |
A knot is a smooth embedding of the circle into the three-sphere S^3. A fundamental operation of producing new knots out of existing ones is called the satellite operation. One such set of satellite knots comes from the so-called Mazur pattern. In general, knots are often studied up to a notion of equivalence, called knot concordance. Two knots are said to be concordant if they jointly form the boundary of an annulus in S^3 x [0,1]. P. Ozsváth and Z. Szabó defined an invariant of the concordance class of a knot, called the tau-invariant. This invariant has been very useful in distinguishing concordance classes of knots.
In 2016, A. Levine used a knot invariant called bordered Floer homology which is well-adapted to studying the satellite construction to give a formula of calculating the tau-invariant of satellite knots with Mazur patterns. In this project, we will generalize Levine's work by considering more general Mazur patterns. These patterns arise from the so-called (1,1)-patterns: patterns that can be drawn on the surface of a torus. We will then give an explicit formula of the tau-invariant for many different families of (1,1)-Mazur patterns. This will involve studying the bordered Heegaard diagrams of (1,1)-knots and the computation of the box tensor product of the bordered Type A and D modules, the work of which is essentially a combinatorial computation.
This week Professor Hendricks and Professor Mallick lectured about the basics of knot theory, knot concordance, the tau invariant, satellite knots, and (bordered) Heegaard Floer homology.
This week we delved deeper into Heegaard Floer homology. We learned about how to construct a Heegaard diagram for knots, and how to construct a filtered chain complex from the diagram. We did some calculations for some simple knots, such as the figure eight knot and the (2,1) cable knot. Then, we studied Levine'e paper calculating the tau invariant for satellite knots with Mazur patterns. This is the main paper we will be working with, which gives the base case for our project. Our first task next week will be to generalize Levine's argument.
This week we began our task in generalizing Levine's argument. We were stuck for a couple of days because we started with the wrong Heegaard diagram (it turns out that there should be always an isotopy such that the alpha and beta arcs have only one algebraic intersection). After a lot of calculations and corrections, we were able to find the tau invariant for one type of general Mazur patterns. This completes our first goal for the project, but we have to be careful and double check!
We also listened to some interesting lectures about Dehn surgery and the Alexander polynomial, and how they relate to the theory of Heegaard Floer homology that we're studying.
This weeek we wrote up our calculations for the tau invariant of our first generalization of the Mazur pattern. Our next goal is to do the same calculation for a different type of generalized Mazur pattern, where we fix the winding number to -+1. By the end of the week, we came up with a stronger theorem, where we calculate the tau invariant for a patterns that wind around the torus m times counterclockwise and n times clockwise, denoted by Q_m,n. The cases when the winding number is -+1 follows immediately when n-m=-+1.
This week we wrote up our calculations for the tau invariant of satellite knots with pattern Q_m,n. We also got started on computing the epsilon invariant for those knots, the base case of which is also exposited in Levine's paper.