My name is Angus Southwell. I am a student studying mathematics and chemistry at Monash University, primarily pure maths. I became interested in maths from reading books as a kid, showing me the interesting side of mathematics beyond what was shown in primary school and high school. Recently I have taken an interest in the mathematical study of knots, which oddly enough is exactly what it sounds like: The mathematics of tied-up bits of string. Despite the simplicity of the objects, knots have interesting algebraic properties and uses. More specifically, my current project studies relationships between polynomials and cord and skein algebras associated with knots. When I’m not getting tangled up in knot theory, I tend to be watching movies or playing video games.
Knots, Cords and Skeins
One of the most basic question of knot theory is to recognise equivalent knots. Two knots are equivalent if one can be rearranged in space, without passing through itself, to obtain the other. Since the number of Reidemeister moves describing these rearrangements have no upper limit, the main tools in recognising whether two diagrams of knots represent equivalent knots are knot invariants: quantities that can be calculated for each knot diagram, but which give the same result on diagrams representing equivalent knots.In the last 30 years, a vast number of knot invariants has been developed, including the Jones polynomial, augmentation polynomial, A-polynomial, Floer homology theories, and many others. This leads to deeper questions such as: What mathematical relationships exist between these structures? This project will investigate some of these relationships. In particular, this project will investigate relationships between the augmentation polynomial AugK(Î»,Âµ,U), the A-polynomial AK(x,y), and the cord and skein algebras of a knot K.The augmentation polynomial AugK(Î»,Âµ,U) originates from the advanced subject of knot contact homology, but it has very recently been shown that AugK(Î»,Âµ,U), can be calculated using some relatively simple matrix algebra. Moreover it is very closely related to an elementary algebra defined by a graphical calculus of cords on the knot, known as the cord algebra.This structure has interesting analogues to the A-polynomial AK(x,y), which is also defined using matrix algebra, and closely related to an elementary graphical algebra known as the skein algebra. In fact, it is known that the A-polynomial essentially divides the augmentation polynomial. In this way, it appears that the augmentation polynomial is a type of generalisation of the A-polynomial, but it is not yet clear precisely in what sense.These developments are very new so there is considerable potential for new observations arising from calculations of augmentation polynomials, and for investigating how the augmentation and A-polynomials are related. As the calculation of augmentation polynomials is fairly elementary, these investigations are well suited to a talented undergraduate student under our supervision.