Grid homology is a combinatorial version of knot Floer homology. In a previous thesis, the author established an unoriented skein exact triangle for grid homology, giving a combinatorial proof of Manolescu’s unoriented skein exact triangle for knot Floer homology, and extending Manolescu’s result from Z/2Z coefficients to coefficients in any commutative ring. In Part II of this dissertation, after recalling the combinatorial proof mentioned above, we track the delta-gradings of the maps involved in the skein exact triangle, and use them to establish the Floer-homological sigma-thinness of quasi-alternating links over any commutative ring. Tangle Floer homology is a combinatorial extension of knot Floer homology to tangles, introduced by Petkova–Vertesi; it assigns an A-infinity-(bi)module to each tangle, so that the knot Floer homology of a link L obtained by gluing together tangles T_1, ..., T_n can be recovered from a tensor product of the A-infinity-(bi)modules assigned to the tangles T_i. Currently, tangle Floer homology has only been defined over Z/2Z. Part III of this dissertation presents a joint result with Ina Petkova, establishing an analogous unoriented skein relation for tangle Floer homology over Z/2Z, and tracking the delta-gradings involved.

Thesis advisor: John Baldwin / Heegaard Floer homology is a family of invariants in low dimensional topology due originally to Ozsváth-Szabó. We discuss various aspects of Heegaard Floer homology and give several link detection results for versions of Heegaard Floer homology for links. In particular, we show that knot and link Floer homology detect various infinite families of cable links. We also give classification results for the Heegaard Floer theoretic invariants of a type of knot called an “almost L-space knot” and an infinite family of detection results for annular Khovanov homology. / Thesis (PhD) — Boston College, 2023. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

In this dissertation, we study the isotopy problem for a certain three-dimensional contactomorphism which is supported in a neighbourhood of an embedded 2-sphere with standard characteristic foliation. The diffeomorphism which underlies it is the Dehn twist on the sphere, and therefore its square becomes smoothly isotopic to the identity. The main result of this dissertation gives conditions under which any iterate of the Dehn twist along a non-trivial sphere is not contact isotopic to the identity. This provides the first examples of exotic contactomorphisms with infinite order in the contact mapping class group, as well as the first examples of exotic contactomorphisms of 3-manifolds with b_1 = 0. The proof crucially relies on the construction of an invariant for families of contact structures in monopole Floer homology which generalises the Kronheimer--Mrowka--Ozsváth--Szabó contact invariant, together with the nice interaction between this families invariant and the U map in Floer homology.

Thesis advisor: John A. Baldwin / Khovanov homology and Heegaard Floer homology have opened new horizons in knot theory and three-manifold topology, respectively. The two invariants have distinct origins, but the Khovanov homology of a link is related to the Heegaard Floer homology of its branched double cover by a spectral sequence constructed by Ozsváth and Szabó. In this thesis, we construct an equivalent spectral sequence with a much more transparent connection to Khovanov homology. This is the first step towards proving Seed and Szabó's conjecture that Szabó's geometric spectral sequence is isomorphic to Ozsváth and Szabó's spectral sequence. These spectral sequences connect information about contact structures contained in each invariant. We construct a braid conjugacy class invariant κ from Khovanov homology by adapting Floer-theoretic tools. There is a related transverse invariant which we conjecture to be effective. The conjugacy class invariant solves the word problem in the braid group among other applications. We have written a computer program to compute the invariant. / Thesis (PhD) — Boston College, 2016. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Kwong, Kwok Kun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 105-109). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgements --- p.iii / Introduction --- p.1 / Chapter 1 --- Morse Theory --- p.4 / Chapter 1.1 --- Introduction --- p.4 / Chapter 1.2 --- Morse Homology --- p.11 / Chapter 2 --- Symplectic Fixed Points and Arnold Conjecture --- p.24 / Chapter 2.1 --- Introduction --- p.24 / Chapter 2.2 --- The Variational Approach --- p.29 / Chapter 2.3 --- Action Functional and Moduli Space --- p.30 / Chapter 2.4 --- Construction of Floer Homology --- p.42 / Chapter 3 --- Fredholm Theory --- p.46 / Chapter 3.1 --- Fredholm Operator --- p.47 / Chapter 3.2 --- The Linearized Operator --- p.48 / Chapter 3.3 --- Maslov Index --- p.50 / Chapter 3.4 --- Fredholm Index --- p.57 / Chapter 4 --- Floer Homology --- p.75 / Chapter 4.1 --- Transversality --- p.75 / Chapter 4.2 --- Compactness and Gluing --- p.76 / Chapter 4.3 --- Floer Homology --- p.88 / Chapter 4.4 --- Invariance of Floer Homology --- p.90 / Chapter 4.5 --- An Isomorphism Theorem --- p.98 / Chapter 4.6 --- Further Applications --- p.103 / Bibliography --- p.105

In 1992, Xiao-Song Lin constructed an invariant h of knots in the 3-sphere via a signed count of the conjugacy classes of irreducible SU(2)-representations of the fundamental group of the knot exterior with trace-free meridians. Lin showed that h equals one-half times the knot signature. Using methods similar to Lin's, we construct an invariant of two-component links in the 3-sphere. Our invariant is a signed count of conjugacy classes of projective SU(2)-representations of the fundamental group of the link exterior with a fixed 2-cocycle and corresponding non-trivial second Stiefel--Whitney class. We show that our invariant is, up to a sign, the linking number. We further construct, for a two-component link in an integral homology sphere, an instanton Floer homology whose Euler characteristic is, up to sign, the linking number between the components of the link. We relate this Floer homology to the Kronheimer-Mrowka instanton Floer homology of knots. We also show that, for two-component links in the 3-sphere, the Floer homology does not vanish unless the link is split.

In this thesis we investigate the problem of defining an extension of sutured instanton Floer homology to give an instanton invariant for a tangle. We do this in three separate steps. First, we investigate the representation variety of singular flat connections on a punctured Riemann surface \(\Sigma\). Suppose \(\Sigma\) has genus \(g\) and that there are \(n\) punctures. We give formulae for the Betti numbers of the space \(\mathcal{R}_{g,n}\) of flat \(SU(2)\)-connections on \(\Sigma\) with trace 0 holonomy around the punctures. By using a natural extension of the Atiyah-Bott generators for the cohomology ring \(H^*(\mathcal{R}_{g,n})\), we are able to write down a presentation for this ring in the case \(g=0\) of a punctured sphere. This is accomplished by studying the intersections of Poincaré dual submanifolds for the new generators and reducing the calculation to a linear algebra problem involving the symplectic volumes of the representation variety. We then study the related problem of computing the instanton Floer homology for a product link in a product 3-manifold <p>\((Y_g, K_n) := (S^1 \times \Sigma, S^1 \times \{n pts\})\).<\p> It is easy to see that the Floer homology of this pair, as a vector space, is essentially the same as the cohomology of \(\mathcal{R}_{g,n}\), and so we set ourselves to determining a presentation for the natural algebra structure on it in the case \(g = 0\). By leveraging a stable parabolic bundles calculation for \(n = 3\) and an easier version of this Floer homology, \(I _*(Y_0, K_n, u)\), we are able to write down a complete presentation for the Floer homology \(I _*(Y_0, K_n)\) as a ring. We recapitulate somewhat the techniques in \([\boldsymbol{27}]\) in order to do this. Crucially, we deduce that the eigenspace for the top eigenvalue for a natural operator \(\mu^{ orb} (\Sigma)\) on \(I_* (Y_0, K_n)\) is 1-dimensional.Finally, we leverage this 1-dimensional eigenspace to define an instanton tangle invariant THI and several variants by mimicking the de nition of sutured Floer homology SHI in \([\boldsymbol{22}]\). We then prove this invariant enjoys nice properties with respect to concatenation, and prove a nontriviality result which shows that it detects the product tangle in certain cases. / Mathematics

This thesis proposes an improved notion of coefficient system for Lagrangian Floer Homology which allows one to produce nontrivial invariants away from characteristic 2, even when coherent orientations of moduli spaces of Floer trajectories do not exist. This explains a suggestion of Witten. The invariant can be computed in examples, and the method explained below should be extensible to other Floer-theoretic invariants. The basic idea is that the moduli spaces of curves admit fundamental classes in homology with coefficients in the orientation lines of the moduli spaces, and the usual construction of coherent orientations actually shows that these fundamental classes naturally map to spaces of paths twisted with appropriate coefficient systems. These twisted path spaces admit enough algebraic structure to make sense of Floer homology with coefficients in these path spaces.

We begin developing a theory of morphisms of moduli spaces of pseudoholomorphic curves and discs with Lagrangian boundary conditions as Kuranishi spaces, using a modification of the procedure of Fukaya-Oh-Ohta-Ono. As an example, we consider the total space of the line bundles O(−n) and O on P1 as toric Kähler manifolds, and we construct isomorphic Kuranishi structures on the moduli space of holomorphic discs in O(−n) on P1 with boundary on a moment map fiber Lagrangian L and on a moduli space of holomorphic discs subject to appropriate tangency conditions in O. We then deform this latter Kuranishi space and use this deformation to define a Lagrangian potential for L in O(−n), and hence a superpotential for O(−n). With some conjectural assumptions regarding scattering diagrams in P1 × P, this superpotential can then be calculated tropically analogously to a bulk-deformed potential of a Lagrangian in P1 × P1.

Dans la première partie de cette thèse, on donne, sous certaines hypothèses, une minoration du nombre de points d’intersections d’une sous-variété Lagrangienne monotone L avec son image par une isotopie Hamiltonienne. Dans le cas où L est un espace K(pi, 1), et en particulier à courbure sectionnelle strictement négative, le minorant est 1 + beta1(L), où beta1 est le premier nombre de Betti à coefficients dans Z2. Une autre conséquence est la non-déplaçabilité d’un plongement Lagrangien monotone de RPn × K (où K est une sous-variété à courbure sectionnelle strictement négative telle que H1(K, Z) ≠ 0) dans certaines variétés symplectiques. Dans la seconde partie, on considère une sous-variété Lagrangienne monotone L non déplaçable. En utilisant l’homologie de Floer définie pour les Lagrangiennes qui sont C-1-proches de L, on obtient des informations sur son nombre de Maslov. De plus, si L peut être approchée par une suite de Lagrangiennes déplaçables, alors, sous certaines hypothèses topologiques sur L, l’énergie de déplacement des éléments de cette suite tend vers l’infini. / N the first part of the thesis, we give, under some hypotheses, a lower bound on the intersection number of a closed monotone Lagrangian submanifold L with its image by a generic Hamiltonianisotopy. For monotone Lagrangian submanifolds L which are K(pi, 1) and, in particular with negative sectional curvature, this bound is 1 + beta_1(L), where beta_1 is the first Betti number with coefficients in Z_2. Another consequence, is the non-displaceability of a monotone Lagrangian embedding of RPn x K (where K is a submanifold with negative sectional curvature such that H^1(K, Z) ≠ 0) in some symplectic manifolds. In the second part, given a closed monotone Lagrangian submanifold L, which is not displaceable, we use Floer homology defined on Lagrangians which are C^1 - close to L, to get information about it Maslov number. Besides, if L can be approached by a sequence of displaceable Lagrangians, then, under some topological assumptions on L, the displacement energy of the elements of this sequence converge to infinity.