Type: Seminar

Modified: Mon 3 Apr 2023 0800; Mon 3 Apr 2023 0807; Mon 3 Apr 2023 0954

Distribution: World

Expiry: 21 Apr 2023

CalTitle1: Pure Honours Talks

Auth: laurent@n58-110-206-239.mas4.nsw.optusnet.com.au (lpau4287) in SMS-SAML

All academic staff, current and prospective Honours students are invited to attend. Thursday 20 April, Carslaw Lecture Theatre 157-257 15:00-15:20 Damian James Lin Title: Virtualising the $d$-invariant of a Knot Abstract: Conway mutation is an operation that transforms knots and knots related in this way are called mutants. The d-invariant, or equivalently, the lattice of integer flows of the Tait graph is an alternating knot invariant that is able to distinguish the mutation classes of alternating knots (Greene 2012). There is an interesting generalisation of knots by Kauffman (1999) and Kuperberg (2003) to surfaces of higher genus, these knots are known as virtual knots. We generalise the d-invariant/lattice of integer flows to alternating virtual knots and prove that in this context, white it remains as a mutation invariant of alternating knots, it is no longer complete up to mutation. We present a counterexample to its completeness by using a different invariant, the Gordon-Litherland linking form to see that there are two non-mutant alternating virtual knots that have the same d-invariant. These knots were found by implementing a program to compute both invariants for virtual knots up to 6 crossings. This talk is based on joint work with Hans Boden, Zsuzsanna Dancso and Tilda Wilkinson. 15:25-15:45 Isaac James Sandeman Green The abstract is: This talk aims to communicate the research undertaken for my honours degree, presented in a style which favours intuitive explanations over formality. It will begin with an introduction to equivariant K-Theory, the core piece of technical machinery used in the project. Then I will discuss how localisation theorems can be used to reduce certain calculations to the moment graph of a space - a combinatoric object. Finally, I will explain how I applied these techniques to the affine Grassmannian of SL2, interpreting the results in the context of the geometric Satake equivalence.