My research
Mathematical interests
A knot is simply a circle sitting in 3-dimensional space. I study knots which are the boundaries of discs in the 4th dimension; such knots are called slice knots. Most knots aren't slice. It is a difficult job to prove that a knot is slice, and often a more difficult job to prove that it isn't! Adding to the complications, it may be that a knot isn't slice, but two times it is slice. Such knots represent elements of finite order in the knot concordance group. It is unknown whether there are any elements of order other than 2 in this group; this is the question I have been addressing in my thesis. I use the techniques of twisted Alexander polynomials and extend the work done by Professors Charles Livingston and Paul Kirk at the University of Indiana.
In the first year of my thesis I worked on an algorithm to compute Seifert matrices from braids. Take a look at the associated website to see the algorithm in action!
Papers and projects
- My PhD thesis is now available to view online!
- My most recent paper finds a formula for the L2 signatures of torus knots and applies this to a theorem of Cochran-Orr-Teichner to prove that the twist knots aren't slice.
- Mark Powell and I have been trying to type up and make sense of Peter Teichner's lecture course on slice knots.
- Some stuff on infinite cyclic covers.
- Some knot theory notes which I made when I started my PhD.
- As part of an undergraduate research project I investigated asymmetric metric spaces with Dr Johannes Zimmer in Bath. This has now been published in Topology and its Applications, Volume 154, Issue 11, pp 2312-2322. It is available online at Science Direct or, for those people without permission to access the website, a preprint of our paper is available on the University of Bath preprint server.
- My final semester masters project was on Homological Algebra.
Conferences past and future
- The Kervaire invariant and stable homotopy theory was held at ICMS (Edinburgh), 25-29th April 2011.
- Maths2010 - the British Mathematical Colloquium and British Applied Mathematical Colloquium were held in in Edinburgh, 6-9 April 2010
- Atiyah80 was held at the University of Edinburgh, 20-22nd April, in honour of Sir Michael Atiyah's 80th birthday.
- Categorification and Geometrisation was held at the University of Glasgow, 13-18th April 2009.
- The Mathematics of Knots: Theory and Application was held at the University of Heidelberg, 15-19th December 2008.
- Surgery and Manifold Theory was held at Münster Universität, 18-20th June 2008 in honour of my supervisor Andrew Ranicki's 60th birthday.
- "Knot Theory: Fifty Years Since Fox and Milnor" was held in Brandeis University, 2nd - 5th June 2008, in memory of Jerry Levine.
- The 22nd "British Topology Meeting" was held at the University of Sheffield, 10-12th September 2007.
- The Program on Braids and Prima Summer School were held at the National University of Singapore, 4-29th June 2007. I was honoured to have been able to give a talk there on the computation of Seifert Matrices.
- The 59th British Mathematical Colloquium was held at Swansea University, 16-19th April 2007.
- Surgery Theory: Past, Present and Future was held at ICMS (Edinburgh), 3-5th July 2006.