Equivariant singular instanton homology, I: Applications to 4D clasp numbers-Aliakbar Daemi (Washington University in St Louis)
Abstract: Any knot K is the boundary of a normally immersed disc in the 4-ball, and the 4D clasp number of K is the smallest number of double points of any such immersed disc. The 4D clasp number of K is bounded below by the slice genus of K. Motivated by the Caporaso-Harris-Mazur conjecture about algebraic curves in a quintic surface, Kronheimer and Mrowka asked whether the difference between the 4D clasp number and the slice genus can be arbitrarily large. In this talk I will introduce a knot invariant, called Gamma, and review some of its properties. Then I explain how this invariant can be used to answer Kronheimer and Mrowka's question. This is joint work with Chris Scaduto.
Equivariant singular instanton homology, II: Introduction to the constructions- Chris Scaduto (University of Miami)
Abstract: The Gamma-invariant introduced in part I of this series is one of several outputs from equivariant singular instanton theory. This framework associates to a knot a suite of invariants which are morally derived from Morse theory of the Chern-Simons functional on an infinite-dimensional space with a circle action. After some general background, I will focus on the algebraic structures that are forefront to the theory. This is joint work with Ali Daemi.
Equivariant singular instanton homology, III: Singular Froyshov invariants and the Gamma-invariant--Aliakbar Daemi (Washington University in St Louis)
Abstract: A homology concordance is an embedded cylinder in a homology cobordism. Equivariant singular instanton homology is functorial with respect to homology concordances. We use this to produce a family of homology concordance invariants. The simplest elements of this family are integer valued invariants which are obtained by imitating the definition of the Froyshov homomorphisms in the context of three manifold invariants. The Gamma invariant used in the first talk is a refinement of these singular Froyshov invariants. This is joint work with Chris Scaduto.
Equivariant singular instanton homology, IV: Further applications- Chris Scaduto (University of Miami)
Abstract: In this final talk I will present additional applications of the theory discussed in the previous parts of the series. One application is that certain topological assumptions imply the existence of non-trivial SU(2) representations for fundamental groups of knotted surface complements. Another application confirms a conjecture of Poudel-Saveliev and computes the irreducible mod 4 graded instanton homology of torus knots. This is joint work with Ali Daemi.
Framed Instanton Floer homology revisited via sutures- Zhenkun Li (MIT)
Abstract: Framed Instanton Floer homology was introduced by Kronheimer and Mrowka for closed oriented 3-manifolds. It is conjectured to be isomorphic to the hat version of Heegaard Floer homology, and recently many computational results were achieved by several groups of people. In this talk, I will explain how the framed Instanton Floer homology of a closed oriented 3-manifold Y can be related to the sutured instanton Floer homology of the complement of a torsion knot inside Y with some suitable sutures. Then several applications follow. This is partially jointed with Sudipta Ghosh and C.-M. Michael Wong.
SL(2,C) Floer Homology, I: The 3-manifold invariant HP(Y)- Ikshu Neithalath (UCLA)
Abstract: We will construct HP(Y), the SL(2,C) Floer homology of a 3-manifold Y, as defined by Abouzaid and Manolescu. To do so, we will give a brief overview of the necessary algebro-geometric tools such as character varieties and perverse sheaves of vanishing cycles.
SL(2,C) Floer Homology, II: Invariants for Knots- Ikshu Neithalath (UCLA)
Abstract: We will sketch the construction of SL(2,C) Floer homology for knots as defined by Cote and Manolescu. We will then discuss some joint work with Cote on the properties of the knot invariant as well as some independent work computing the 3-manifold invariant for surgeries on knots.