Floer Homology Research Articles

Corrigendum: A flexible construction of equivariant Floer homology and applications

We correct a mistake regarding almost complex structures on Hilbert schemes of points in surfaces in arXiv:1510.02449. The error does not affect the main results of the paper, and only affects the proofs of invariance of equivariant symplectic Khovanov homology and reduced symplectic Khovanov homology. We give an alternate proof of the invariance of equivariant symplectic Khovanov homology.

Connected Floer homology of covering involutions

Using the covering involution on the double branched cover of S^3 branched along a knot, and adapting ideas of Hendricks–Manolescu and Hendricks–Hom–Lidman, we define new knot (concordance) invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots.

On spectral sequences from Khovanov homology

There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary 1-handle movie moves induce a morphism of spectral sequences. These morphisms remain unexploited in the literature, perhaps because there is still an open question concerning the naturality of maps induced by general movies. In this paper we focus on the spectral sequences due to Kronheimer-Mrowka from Khovanov homology to instanton knot Floer homology, and on that due to Ozsv\'ath-Szab\'o to the Heegaard-Floer homology of the branched double cover. For example, we use the 1-handle morphisms to give new information about the filtrations on the instanton knot Floer homology of the (4,5) torus knot, determining these up to an ambiguity in a pair of degrees; to determine the Ozsv\'ath-Szab\'o spectral sequence for an infinite class of prime knots; and to show that higher differentials of both the Kronheimer-Mrowka and the Ozsv\'ath-Szab\'o spectral sequences necessarily lower the delta grading for all pretzel knots.

GENERATORS, RELATIONS, AND HOMOLOGY FOR OZSVÁTH–SZABÓ’S KAUFFMAN-STATES ALGEBRAS

We give a generators-and-relations description of differential graded algebras recently introduced by Ozsváth and Szabó for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.

An introduction to knot Floer homology and curved bordered algebras

We survey Ozsvath–Szabo’s bordered approach to knot Floer homology. After a quick introduction to knot Floer homology, we introduce the relevant algebraic concepts ($$\mathcal {A}_\infty $$-modules, type D-structures, box tensor product, etc.), we discuss partial Kauffman states, the construction of the boundary algebra, and sketch Ozsvath and Szabo’s analytic construction of the type D-structure associated to an upper diagram. Finally we give an explicit description of the structure maps of the DA-bimodules of some elementary partial diagrams. These can be used to perform explicit computations of the knot Floer differential of any knot in $$S^3$$. The boundary DGAs $$\mathcal {B}(n,k)$$ and $$\mathcal {A}(n,k)$$ of Ozsvath and Szabo (‘Kauffman states, bordered algebras, and a bigraded knot invariant’, 2016. arXiv:1603.06559) are replaced here by an associative algebra $$\mathcal {C}(n)$$. These are the notes of two lecture series delivered by Peter Ozsvath and Zoltan Szabo at Princeton University during the summer of 2018.

Sutured manifolds and polynomial invariants from higher rank bundles

For each integer $N\geq 2$, Mari\~no and Moore defined generalized Donaldson invariants by the methods of quantum field theory, and made predictions about the values of these invariants. Subsequently, Kronheimer gave a rigorous definition of generalized Donaldson invariants using the moduli spaces of anti-self-dual connections on hermitian vector bundles of rank $N$. In this paper, Mari\~no and Moore's predictions are confirmed for simply connected elliptic surfaces without multiple fibers and certain surfaces of general type in the case that $N=3$. The primary motivation is to study 3-manifold instanton Floer homologies which are defined by higher rank bundles. In particular, the computation of the generalized Donaldson invariants are exploited to define a Floer homology theory for sutured 3-manifolds.

Contact handles, duality, and sutured Floer homology

We give an explicit construction of the Honda--Kazez--Mati\'c gluing maps in terms of contact handles. We use this to prove a duality result for turning a sutured manifold cobordism around, and to compute the trace in the sutured Floer TQFT. We also show that the decorated link cobordism maps on the hat version of link Floer homology defined by the first author via sutured manifold cobordisms and by the second author via elementary cobordisms agree.

Legendrian DGA Representations and the Colored Kauffman Polynomial

For any Legendrian knot $K$ in standard contact ${\mathbb R}^3$ we relate counts of ungraded ($1$-graded) representations of the Legendrian contact homology DG-algebra $(\mathcal{A}(K),\partial)$ with the $n$-colored Kauffman polynomial. To do this, we introduce an ungraded $n$-colored ruling polynomial, $R^1_{n,K}(q)$, as a linear combination of reduced ruling polynomials of positive permutation braids and show that (i) $R^1_{n,K}(q)$ arises as a specialization $F_{n,K}(a,q)\big|_{a^{-1}=0}$ of the $n$-colored Kauffman polynomial and (ii) when $q$ is a power of two $R^1_{n,K}(q)$ agrees with the total ungraded representation number, $\operatorname{Rep}_1\big(K, \mathbb{F}_q^n\big)$, which is a normalized count of $n$-dimensional representations of $(\mathcal{A}(K),\partial)$ over the finite field $\mathbb{F}_q$. This complements results from [Leverson C., Rutherford D., Quantum Topol. 11 (2020), 55-118, arXiv:1802.10531] concerning the colored HOMFLY-PT polynomial, $m$-graded representation numbers, and $m$-graded ruling polynomials with $m \neq 1$.

Surgery, polygons and SU(N)‐Floer homology

Surgery exact triangles in various 3-manifold Floer homology theories provide an important tool in studying and computing the relevant Floer homology groups. These exact triangles relate the invariants of 3-manifolds, obtained by three different Dehn surgeries on a fixed knot. In this paper, the behavior of $SU(N)$-instanton Floer homology with respect to Dehn surgery is studied. In particular, it is shown that there are surgery exact tetragons and pentagons, respectively, for $SU(3)$- and $SU(4)$-instanton Floer homologies. It is also conjectured that $SU(N)$-instanton Floer homology in general admits a surgery exact $(N+1)$-gon. An essential step in the proof is the construction of a family of asymptotically cylindrical metrics on ALE spaces of type $A_n$. This family is parametrized by the $(n-2)$-dimensional associahedron and consists of anti-self-dual metrics with positive scalar curvature. The metrics in the family also admit a torus symmetry.

A Chekanov–Eliashberg algebra for Legendrian graphs

We define a differential graded algebra for Legendrian graphs and tangles in the standard contact Euclidean three space. This invariant is defined combinatorially by using ideas from Legendrian contact homology. The construction is distinguished from other versions of Legendrian contact algebra by the vertices of Legendrian graphs. A set of countably many generators and a generalized notion of equivalence are introduced for invariance. We show a van Kampen type theorem for the differential graded algebras under the tangle replacement. Our construction recovers many known algebraic constructions of Legendrian links via suitable operations at the vertices.

Tight contact structures on Seifert surface complements

We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of the contact structures are identified with hypertrees in a certain hypergraph. Using earlier work, this establishes a connection between contact topology and the Homfly polynomial. We also show that the contact invariants of our tight contact structures form a basis for sutured Floer homology. Finally, we relate our methods and results to Kauffman's formal knot theory.

Signature invariants related to the unknotting number

New lower bounds on the unknotting number of a knot are constructed from the classical knot signature function. These bounds can be twice as strong as previously known signature bounds. They can also be stronger than known bounds arising from Heegaard Floer and Khovanov homology. Results include new bounds on the Gordian distance between knots and information about four-dimensional knot invariants. By considering a related non-balanced signature function, bounds on the unknotting number of slice knots are constructed; these are related to the property of double-sliceness.

The knot Floer cube of resolutions and the composition product

We examine the relationship between the oriented cube of resolutions for knot Floer homology and HOMFLY-PT homology. By using a filtration induced by additional basepoints on the Heegaard diagram for a knot [Formula: see text], we see that the filtered complex decomposes as a direct sum of HOMFLY-PT complexes of various subdiagrams. Applying Jaeger’s composition product formula for knot polynomials, we deduce that the graded Euler characteristic of this direct sum is the HOMFLY-PT polynomial of [Formula: see text].

Indefinite Stein fillings and $$\text {PIN}(2)$$-monopole Floer homology

We introduce techniques to study the topology of Stein fillings of a given contact three-manifold $$(Y,\xi )$$ which are not negative definite. For example, given a $$\hbox {spin}^c$$ rational homology sphere $$(Y,{\mathfrak {s}})$$ with $${\mathfrak {s}}$$ self-conjugate such that the reduced monopole Floer homology group $${\textit{HM}}_{\bullet }(Y,{\mathfrak {s}})$$ has dimension one, we show that any Stein filling which is not negative definite has $$b_2^+=1$$ or 2, and $$b_2^-$$ is determined in terms of the Froyshov invariant. The proof of this uses $$\text {Pin}(2)$$-monopole Floer homology. More generally, we prove that analogous statements hold under certain assumptions on the contact invariant of $$\xi $$ and its interaction with $$\text {Pin}(2)$$-symmetry. We also discuss consequences for finiteness questions about Stein fillings.

Skein relations for tangle Floer homology

In a previous paper, Vértesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle T a differential graded bimodule \tilde{\mathrm {CT}} (T) . If L is obtained by gluing together T_1, \cdots, T_m , then the knot Floer homology \widehat{\mathrm {HFK}} (L) of L can be recovered from \widetilde{\mathrm {CT}}(T_1), \cdots, \widetilde{\mathrm{CT}} (T_m) . In the present paper, we prove combinatorially that tangle Floer homology satisfies unoriented and oriented skein relations, which are analogues of the skein exact triangles for knot Floer homology.

Monotone Lagrangians in of minimal Maslov number n + 1

AbstractWe show that a monotone Lagrangian L in ${\mathbb{C}}{\mathbb{P}}^n$ of minimal Maslov number n + 1 is homeomorphic to a double quotient of a sphere, and thus homotopy equivalent to ${\mathbb{R}}{\mathbb{P}}^n$. To prove this we use Zapolsky’s canonical pearl complex for L over ${\mathbb{Z}}$, and twisted versions thereof, where the twisting is determined by connected covers of L. The main tool is the action of the quantum cohomology of ${\mathbb{C}}{\mathbb{P}}^n$ on the resulting Floer homologies.

A simple construction of an action selector on aspherical symplectic manifolds

We construct an action selector on aspherical symplectic manifolds that are closed or convex. Such selectors have been constructed by Matthias Schwarz using Floer homology. The construction we present here is simpler and uses only Gromov compactness.

A brief survey of the spectral numbers in floer homology

We give a brief introduction and a partial survey of some of the results about spectral numbers in symplectic topology that we are aware of. Without attempting to be comprehensive, we will select just some of the constructions and ideas that, according to our personal taste and our point of view, give a flavor of this fast developing theory.

Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials

We sketch a construction of Legendrian Symplectic Field Theory (SFT) for conormal tori of knots and links. Using large $N$ duality and Witten's connection between open Gromov-Witten invariants and Chern-Simons gauge theory, we relate the SFT of a link conormal to the colored HOMFLY-PT polynomials of the link. We present an argument that the HOMFLY-PT wave function is determined from SFT by induction on Euler characteristic, and also show how to, more directly, extract its recursion relation by elimination theory applied to finitely many noncommutative equations. The latter can be viewed as the higher genus counterpart of the relation between the augmentation variety and Gromov-Witten disk potentials established by Aganagic, Vafa, and the authors, and, from this perspective, our results can be seen as an SFT approach to quantizing the augmentation variety.

Non-integer characterizing slopes for torus knots

A slope $p/q$ is a characterizing slope for a knot $K$ in $S^3$ if the oriented homeomorphism type of $p/q$-surgery on $K$ determines $K$ uniquely. We show that for each torus knot its set of characterizing slopes contains all but finitely many non-integer slopes. This generalizes work of Ni and Zhang who established such a result for $T_{5,2}$. Along the way we show that if two knots $K$ and $K'$ in $S^3$ have homeomorphic $p/q$-surgeries, then for $q\geq 3$ and $p$ sufficiently large we can conclude that $K$ and $K'$ have the same genera and Alexander polynomials. This is achieved by consideration of the absolute grading on Heegaard Floer homology.