Alternatively have a look at the program.

## Holonomy perturbations in instanton gauge theory and irreducible SL(2,C)-representations of integer homology spheres I

We use the technique of holonomy perturbations in instanton gauge theory, which can be traced back

to Floer and Taubes, to prove some topological results about the image of the representation variety

of the complement of a non-trivial knot in the representation variety of its boundary torus.

The first part discusses some of our recent results and how these imply that integer homology 3-spheres

different from the 3-sphere admit irreducible representations of their fundamental group in SL(2,C),

## Holonomy perturbations in instanton gauge theory and irreducible SL(2,C)-representations of integer homology spheres II

The second part discusses in more detail the actual role played by holonomy perturbations in the proofs.

In fact, the holonomy-perturbed flat connections interfere with the representation variety of the knot

complement, and in a more flexible way than one may have thought. The essential non-triviality results

for the space of these perturbed connections come from the TQFT-properties of instanton gauge theory,

and from a theorem of Kronheimer-Mrowka that states that the 0-surgery on a knot embeds as a splitting

## SU(2)-cyclic surgeries and the pillowcase

A 3-manifold is called SU(2)-cyclic if every homomorphism from its

fundamental group to SU(2) has cyclic image. In this talk, we will

study the question of when knots in S 3 have infinitely many

SU(2)-cyclic surgeries. We will show that for any such knot, the set

of SU(2)-cyclic surgery slopes has a single limit point, and then by

applying holonomy perturbations to Kronheimer and Mrowka’s instanton

knot homology, we will show that this limit point is finite. We thus

## Khovanov homology detects the trefoils

In 2010, Kronheimer and Mrowka proved that Khovanov homology detects the unknot, by constructing a spectral sequence from reduced Khovanov homology to a form of instanton knot homology and then showing that the latter has rank 1 only for the unknot. In this talk, we will discuss joint work with John Baldwin in which we apply techniques from contact geometry to prove that the rank of instanton knot homology also detects the trefoils. As a corollary, we conclude that Khovanov homology is also a trefoil detector.

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