**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.

## 11 replies on “Floer theory and low-dimensional topology: Gauge theory”

[…] Gauge theory: […]

Hello, my name is Ikshu Neithalath. I’m a graduate student at UCLA (currently visiting at Stanford) under Ciprian Manolescu. My talks about SL(2,C) Floer homology are a bit different from the other talks in this mini-session because they don’t actually involve any gauge theory! Gauge theory only serves as a motivation and guiding tool for developing SL(2,C) Floer homology, but the methods we’ll be using come from algebraic geometry.

I apologize for any typos (text or verbal) in my talks. Thanks to the magic of editing, I caught several in production but I assume some persisted. Normally you’d be able to clarify with me immediately, but in this format you’ll need to ask me a question here.

I welcome all questions and will try to reply in a timely manner throughout the conference. If you’d prefer to ask me in person, then please attend my offices hour at 3:30 on Friday, June 5. I would like to keep my office hour mainly for graduate students, but of course anyone is welcome.

(Edit: My office hours are at 4:30 on Friday, June 5. See the schedule for the correct information)

Hi Ali and Chris–Very nice talks (so far–I’ve watched the first two). In Chris’s talk II, a local system $\Delta$ was discussed, but (unless I missed something) not defined. Can you say what the representation of $\pi_1(B)$ is that defines this local system?

Chris replied off-line (due to some funkiness with registering). Here’s his answer:

The local system is as follows. The configuration space of singular SU(2) connections for a knot has two natural functionals to the circle. One is the (singular) Chern-Simons functional. The other roughly takes the holonomy around the longitudinal direction of the knot. (This is how Kronheimer and Mrowka define their local systems.) Then the local system is basically the pull back of a standard local system on $S^1\times S^1$ by the product of these functionals.

Thanks Danny and Chris for posting the answer.

Hi Ikshu, I enjoyed your talks! Can you define a tau-weighted sheaf-theoretic SL(2,C) Floer cohomologies HP for links — perhaps, using the knottification method of turning a link in $Y$ into a knot in a connected sum of copies of $S^1 \times S^2$? Also, is it expected (or known) that cobordisms between three-manifolds induce maps between their sheaf-theoretic SL(2,C) homologies?

Linh,

Thank you for your questions. A priori, I don’t know if the construction used to define HP can be adapted to the case of links. I’ll have to think a bit more about that. But since HP_tau is defined for knots in any closed, connected, orientable 3-manifold, taking HP_tau of the knottification should indeed produce a link invariant (at least for knots if S^3, my understanding is that knottification is defined only for knots in S^3 but that might be a misconception).

The TQFT properties of SL(2,C) Floer homology are not known, but are expected to exist. The theory should be functorial under cobordisms as part of a 3+1 dimensional TQFT. Here’s the basic setup: Let W be a cobordism from Y_1 to Y_2. Then on character varieties we get a map X(W)-> X(Y_1)\times X(Y_2). This should be a Lagrangian morphism to a -1 shifted symplectic scheme. We want to think of it as a Lagrangian correspondence between the two -1 shifted symplectic schemes X(Y_1) and X(Y_2). Does such a correspondence give a map between the canonical perverse sheaves on the X(Y_i)? My understanding is that the algebraic geometry to prove this statement has yet to be done (at least that was the status when I last thought about this). See Conjecture 5.22 in “Perversely Categorified Lagrangian Correspondences” by Amorim and Ben-Bassat. I think that conjecture is the main input we’d need to define the cobordism maps.

Hi Zhenkun, I enjoyed your talk! Are there examples of knots $K$ in $S^3$ (or other three-manifolds $Y$) for which your rank inequality $dim KHI(Y,K) \geq dim I^\#(Y)$ is sharp? Thanks!

Hi Ikshu,

I really like this result for small knots. Is there hope for some detection results going in the opposite way, i.e. HP detecting if a knot is small, or more generally, the maximal dimension of a component of the character variety? Or can the homology “disappear” if the variety is too singular?

Hi Tye,

Thanks for your question. Just as a reminder for context, Culler-Shalen theory says that if K is a small knot, then the character variety of its exterior is 1-dimensional. Also, the character variety of any knot exterior is always positive dimensional. See 2.4 in “Plane Curves Associated to Character Varieties of 3-Manifolds” by Cooper, Culler, Gillet, Long and Shalen. This is the paper where they define the A-polynomial.

Your question was about a possible converse to this result. Conjecture 1: If K is a large knot, then the maximal dimension of a component of its character variety is greater than 1. As far as I know, this conjecture is open.

Even if this were resolved, your second question about HP still stands. We can consider Conjecture 2: If HP(Y_r(K)) is concentrated in degree zero for most surgery coefficients r, then K is small. Certainly if Conjecture 1 is false then so is Conjecture 2.

However, Conjecture 1 is not the only obstruction to Conjecture 2. One would also need to know something about local systems being trivial. For example, it is possible that the character variety of one our surgeries contains a smooth copy of C^*. If our perverse sheaf were just the trivial local system with stalk Z in degree -1, then there would be cohomology in degree -1. But if it were the non-trivial local system (where the monodromy acts by -1 on Z), then the cohomology would be concentrated in degree 0. Perhaps there is even a more sophisticated example of a local system in which the integral cohomology vanishes in all degrees. As of yet, we do not have techniques to rule out such situations involving non-trivial local systems.