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Abstracts for Informal mini-workshop: A celebration of instantons

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

Posted in
Speaker: 
Raphael Zentner
Zugehörigkeit: 
Universität Regensburg/MPI
Datum: 
Die, 25/07/2017 - 10:00 - 10:55
Location: 
MPIM Seminar Room
Parent event: 
Informal mini-workshop: A celebration of instantons

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

Posted in
Speaker: 
Raphael Zentner
Zugehörigkeit: 
Universität Regensburg/MPI
Datum: 
Die, 25/07/2017 - 11:05 - 12:00
Location: 
MPIM Seminar Room
Parent event: 
Informal mini-workshop: A celebration of instantons

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

Posted in
Speaker: 
Steven Sivek
Zugehörigkeit: 
Princeton University/MPI
Datum: 
Die, 25/07/2017 - 14:00 - 14:55
Location: 
MPIM Seminar Room
Parent event: 
Informal mini-workshop: A celebration of instantons

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

Posted in
Speaker: 
Steven Sivek
Zugehörigkeit: 
Princeton University/MPI
Datum: 
Die, 25/07/2017 - 15:05 - 16:00
Location: 
MPIM Seminar Room
Parent event: 
Informal mini-workshop: A celebration of instantons 
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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