Knot Days
Virtual summer school, 15-19 November 2021
All Australian HDR students are invited to join us. Please use the link below to access the registration page:
The workshop will have three mini-classes, each with lectures and problem sessions.
(1) Introduction to Legendrian Knot Theory (Joan Licata, ANU)
This class will be broadly accessible, requiring no background in topology. We'll introduce the basics of general knot theory (diagrams, Reidemeister moves, invariants) alongside features specific to the knot theory in contact three-manifolds.
(2) Character Varieties, A-polynomials and Knots (Stephan Tillmann, Sydney)
Many properties that allow us to distinguish and study knots are not properties of the knot, but rather of the complement of a knot. This three-dimensional space may appear less tangible than the actual knot, but allows the definition of algebraic invariants that encode information about the knot and its complement. This series of lectures focuses on invariants arising from algebraic geometry. These can be used to detect interesting surfaces spanned by knots, to recognise whether a knot is in fact knotted, and to determine whether a knot complement has a geometric structure of constant negative curvature.
These lectures will provide an overview over the main aspects of what is broadly known as Culler-Shalen theory, and describe some key applications. The techniques mix ideas from group theory, algebraic geometry and geometric topology. The level of detail given will depend on the background and interest of the audience.
Supporting material for the lectures is an old set of notes (with references to the original literature):
https://www.maths.usyd.edu.au/u/tillmann/papers/course.pdf
A rough plan for the lectures is:
Lecture 1 (Monday 10:45): Motivation, some hyperbolic geometry, the canonical example
Lecture 2 (Wednesday 9:30): Surfaces and actions on trees (Sections 2 & 3)
Lecture 3 (Thursday 10:30): Surfaces associated to ideal points (Sections 4 & 5)
Lecture 4 (Friday 10:30): The Weak Neuwirth Conjecture and the Roots of Unity Phenomenon (Sections 6, 7, and 9)
The notes contain details and material not covered in lectures, as well as exercises.
(3) Jones Polynomial and Volume Conjectures (Dan Mathews, Monash)
Knots can be studied from some very different perspectives, but there are some deep conjectures that unify these perspectives. In this series of lectures we will discuss some of these different perspectives and two of the major conjectures connecting them: the volume conjecture and the AJ conjecture.
Starting from the Jones polynomial, we'll give an overview of the broad range of ideas around these conjectures, including coloured Jones polynomials, quantum invariants, q-holonomicity, hyperbolic geometry, and skein algebras. No background will be assumed, but some knowledge of abstract algebra will be useful.
Information for Participants:
Please keep an eye out for an email with Zoom details coming soon.
All participants are invited to attend the first three lectures on Monday. For the rest of the week, you should choose one or two classes, but not all three ---too much Zoom!
Monday: 9:30 Welcome, 9:35 Licata lecture, 10:20 BYO tea, 10:45 Tillmann lecture, 1:00 Mathews lecture , 2:00 Licata problem session
Tuesday morning: 9:30 Mathews problem session, 11:00 Licata lecture 1:00 Tillmann problem session, 2:15 Mathews lecture
Wednesday morning: 9:30 Tillmann lecture, 11 Licata problem session 1:00-3:00 joint Tillmann-Mathews problem session
Thursday: 9:30 Licata lecture, 10:30 Tillmann lecture, 1:00 Mathews lecture, 2:15 joint problem sessions
Friday: 9:30 Licata lecture, 10:30 Tillmann lecture, 1:00 Mathews lecture
Participants are encouraged to join with video and ask lots of questions!
Continuation of the Character Varieties course (the ZOOM link remains the same):
Wednesday, 24 November, 10:00-11:30
Wednesday, 1 December, 9:30-10:30