Pure Maths Honours Talks -- Honours Talks

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.