# TTT105: Sheffield

The University of Sheffield.

Supported by the London Mathematical Society.

## Date

24 January 2018

## Speakers

### Will Rushworth (Durham)

Doubled Khovanov homology

Doubled Khovanov homology

Abstract: Virtual knot theory is an extension of classical knot theory which considers knots and links in equivalence classes of thickened orientable surfaces. Khovanov homology is a powerful invariant of classical links, and it can be applied to virtual links using Z_2 coefficients. However, a number of problems arise when one attempts to use other coefficient rings.

In this talk we describe doubled Khovanov homology: an extension of Khovanov homology to virtual links with arbitrary coefficients. Unlike other extensions of Khovanov homology, doubled Khovanov homology requires no new diagrammatics, as all the work is done algebraically. We shall describe the construction of the invariant as well as some of its applications, in particular to virtual knot concordance.

### Jocelyne Ishak (Kent)

Rigidity of the K(1)-local stable homotopy category

Rigidity of the K(1)-local stable homotopy category

Abstract: In some cases, it is sufficient to work in the homotopy category Ho(C) associated to a model category C, but looking at the homotopy level alone does not provide us with higher order structure information. Therefore, we investigate the question of rigidity: if we just had the structure of the homotopy category, how much of the underlying model structure can we recover?

For example, the stable homotopy category Ho(Sp) has been proved to be rigid by S Schwede. Moreover, the E(1)-local stable homotopy category, for p=2, has been shown to be rigid by C Roitzheim.

In this talk, I will discuss a new case of rigidity, which is the localisation of spectra with respect to the Morava K-theory K(1), at p=2. While the K(n)-local spectra can be related to the E(n)-local spectra, there are a lot of main differences to keep in mind while studying the rigidity in the K(1)-local case.

Therefore, what might be true and applicable for the E(1)-localisation studied by C Roitzheim might not be true anymore in the K(1)-local world. In this talk, I will emphasis those differences, and sketch the proof of the rigidity of the K(1)-local stable homotopy category at p=2.

### Richard Webb (Cambridge)

The conjugacy problem for the mapping class group in polynomial time

The conjugacy problem for the mapping class group in polynomial time

Abstract: Fix an orientable surface S of finite type. We shall describe an algorithm that solves the conjugacy problem in the mapping class group of S in polynomial time. This includes closed surfaces S and the braid groups. Joint work with Mark Bell.