68 pages, 31 figures; v3 to appear in Quantum Topology; v2 substantially revised from v1 including weakened statement of main result due to error in v1's Lemma 5 (conjugation / moving the basepoint). Other changes: substantial elaboration of argument for Reidemeister 3 invariance, different descriptions of behavior under stabilization and adding/removing layers of marked points
This paper considers the invariance of knot Floer homology in a purely algebraic setting, without reference to Heegaard diagrams, holomorphic disks, or grid diagrams. We show that (a small modification of) Ozsváth and Szabó's cube of resolutions for knot Floer homology, which is assigned to a braid presentation with a basepoint, is invariant under braid-like Reidemeister moves II and III and under conjugation. All moves are assumed to happen away from the basepoint. We also describe the behavior of the cube of resolutions under stabilization. The techniques echo those employed to prove the invariance of HOMFLY-PT homology by Khovanov and Rozansky, and are further evidence of a close relationship between the theories. The key idea is to prove categorified versions of certain equalities satisfied by the Murakami-Ohtsuki-Yamada state model for the HOMFLY-PT polynomial.
This is a survey article about knot Floer homology. We present three constructions of this invariant: the original one using holomorphic disks, a combinatorial description using grid diagrams, and a combinatorial description in terms of the cube of resolutions. We discuss the geometric information carried by knot Floer homology, and the connection to three- and four-dimensional topology via surgery formulas. We also describe some conjectural relations to Khovanov-Rozansky homology.
In this paper, we compute the symplectic Floer homology of the figure eight knot. This provides first nontrivial knot with trivial symplectic Floer homology.
We prove that the rank of knot Floer homology detects the Hopf links, and generalize this result further to classify the links of the second smallest knot Floer homology. We also prove a knot Floer homology analog of arXiv:1910.04246v1 [math.GT] and give a partial answer to when the equality holds in the rank inequality between the knot Floer homology of a link and its sublinks.
We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology. Each knot in this infinite family admits a nontrivial genus two mutant which shares the same total dimension in both knot Floer homology and Khovanov homology. Each knot is distinguished from its genus two mutant by both knot Floer homology and Khovanov homology as bigraded groups. Additionally, for both knot Heegaard Floer homology and Khovanov homology, the genus two mutation interchanges the groups in $δ$-gradings $k$ and $-k$.
The knot Floer complex and the concordance invariant $\varepsilon$ can be used to define a filtration on the smooth concordance group. We exhibit an ordered subset of this filtration that is isomorphic to $\mathbb{N} \times \mathbb{N}$ and consists of topologically slice knots.
This article addresses the two significant aspects of Ozsváth and Szabó's knot Floer cube of resolutions that differentiate it from Khovanov and Rozansky's HOMFLY-PT chain complex: (1) the use of twisted coefficients and (2) the appearance of a mysterious non-local ideal. Our goal is to facilitate progress on Rasmussen's conjecture that a spectral sequence relates the two knot homologies. We replace the language of twisted coefficients with the more quantum topological language of framings on trivalent graphs. We define a homology theory for framed trivalent graphs with boundary that -- for a …
It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of $K$ detects more structure of minimal genus Seifert surfaces for $K$. We define an invariant of algebraically slice, genus one knots and provide examples to show that knot Floer homology does not detect this invariant. Finally, we remark that certain metabelian $L^2$-signatures bound this invariant from below.
We define Floer homology theories for oriented, singular knots in S^3 and show that one of these theories can be defined combinatorially for planar singular knots.
We present a combinatorial method for a calculation of knot Floer homology with Z-coefficient of (1,1)-knots, and then demonstrate it for non-alternating (1,1)-knots with ten crossings and the pretzel knots of type (-2,m,n). Our calculations determine the unknotting numbers and 4-genera of the pretzel knots of this type.
For a knot K and its knot Floer complex CFK^-(K), we introduce an algorithm to compute the bordered Floer bimodule of the complement of the knot and its meridian. The grading of the module computes spin^c-summands of a meridional knot in the large Dehn surgery manifold, which can be also extended to arbitrary framing n.
In this paper, we introduce a sequence of invariants of a knot K in S^3: the knot Floer homology groups of the preimage of K in the m-fold cyclic branched cover over K. We exhibit the knot Floer homology in the m-fold branched cover as the categorification of a multiple of the Turaev torsion in the case where the m-fold branched cover is a rational homology sphere. In addition, when K is a 2-bridge knot, we prove that the knot Floer homology of the lifted knot in a particular Spin^c structure in the branched double cover matches the knot Floer homology of the original knot K in S^3. We conclud…
We show that if a positive integral surgery on a knot K inside a homology sphere X with Seifert genus g(K) results in an induced knot K_n in X_n(K)=Y which has simple Floer homology, we should have n>=2g(K). Moreover, if X is the standard sphere, the three-manifold Y is a L-space and the Heegaard Floer homology groups of K are determined by its Alexander polynomial.
Knot Floer homology is an invariant for knots in the three-sphere for which the Euler characteristic is the Alexander-Conway polynomial of the knot. The aim of this paper is to study this homology for a class of satellite knots, so as to see how a certain relation between the Alexander-Conway polynomials of the satellite, companion and pattern is generalized on the level of the knot Floer homology. We also use our observations to study a classical geometric invariant, the Seifert genus, of our satellite knots.
This note explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology which do not arise as the invariant of a knot. This leads to a new constraint for knots admitting lens space surgeries, as well as a proof that the rank of knot Floer homology detects the trefoil knot. For the second, we show that any non-trivial band sum of two unknots gives rise to…
Let K in S^3 be a knot, and let \widetilde{K} denote the preimage of K inside its double branched cover, Σ(K). We prove, for each integer n > 1, the existence of a spectral sequence from Khovanov's categorification of the reduced n-colored Jones polynomial of the mirror of K to the knot Floer homology of (Σ(K),\widetilde{K}) (when n odd) and to (S^3, K # K) (when n even). A corollary of our result is that Khovanov's categorification of the reduced n-colored Jones polynomial detects the unknot whenever n>1.
Given an equivariant knot $K$ of order $2$, we study the induced action of the symmetry on the knot Floer homology. We relate this action with the induced action of the symmetry on the Heegaard Floer homology of large surgeries on $K$. This surgery formula can be thought of as an equivariant analog of the involutive large surgery formula proved by Hendricks and Manolescu. As a consequence, we obtain that for certain double branched covers of $S^{3}$ and corks, the induced action of the involution on Heegaard Floer homology can be identified with an action on the knot Floer homology. As an appl…
Let $\widetilde{K}$ be a 2-periodic knot in $S^3$ with quotient $K$. We prove a rank inequality between the knot Floer homology of $\widetilde{K}$ and the knot Floer homology of $K$ using a spectral sequence of Hendricks, Lipshitz and Sarkar. We also conjecture a filtered refinement of this inequality, for which we give computational evidence, and produce applications to the Alexander polynomials of $\widetilde{K}$ and $K$.
We review some recent results in knot concordance and homology cobordism. The proofs rely on various forms of Heegaard Floer homology. We also discuss related open problems.
A well-known conjecture of Rasmussen states that for any knot $K$ in $S^{3}$, the rank of the reduced Khovanov homology of $K$ is greater than or equal to the rank of the reduced knot Floer homology of $K$. This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Floer homology. Using an oriented cube of resolutions construction for a homology theory related to knot Floer homology, we prove this conjecture.
We extend an approach of Beliakova for computing knot Floer homology and implement it in a publicly available computer program. We review the main programming and optimization methods used. Our program is then used to check that the Floer homology of a prime non-alternating knot with less than 12 crossings has no torsion.
In this paper we study the knot Floer homology of a subfamily of twisted $(p, q)$ torus knots where $q \equiv\pm1$ (mod $p$). Specifically, we classify the knots in this subfamily that admit L-space surgeries. To do calculations, we use the fact that these knots are $(1, 1)$ knots and, therefore, admit a genus one Heegaard diagram.
We develop a method of calculation for the symplectic Floer homology of composite knots. The symplectic Floer homology of knots defined in \cite{li} naturally admits an integer graded lifting, and it formulates a filtration and induced spectral sequence. Such a spectral sequence converges to the symplectic homology of knots in \cite{li}. We show that there is another spectral sequence which converges to the $\Z$-graded symplectic Floer homology for composite knots represented by braids.
We develop a skein exact sequence for knot Floer homology, involving singular knots. This leads to an explicit, algebraic description of knot Floer homology in terms of a braid projection of the knot.
Using a Heegaard diagram for the pullback of a knot $K \subset S^3$ in its cyclic branched cover $Σ_m(K)$ obtained from a grid diagram for $K$, we give a combinatorial proof for the invariance of the associated combinatorial knot Floer homology over $\mathbb{Z}$.
We prove that the Knot Floer homology group of a fibred knot of genus g in the Alexander grading 1-g is isomorphic to a version of the fixed point Floer homology of an area-preserving representative of the monodromy.
A companion paper to "On knot Floer homology in branched double covers" applied to braided branched loci. We reprove the main result of that paper concerning alternating branched loci when projected to an annulus, without using Khovanov homology. This provides two advantages: 1) the results hold for integer coefficients and 2) the spin^c structures are more readily discernable. We apply this result to a branch locus which is a braid, and use the braid structure to find information about a fibered knot in the branched double cover. In some cases this provides all the information about the knot …
We define a new smooth concordance homomorphism based on the knot Floer complex and an associated concordance invariant, epsilon. As an application, we show that an infinite family of topologically slice knots are independent in the smooth concordance group.
We prove that, up to local equivalences, a suitable truncation of the involutive knot Floer homology of a knot in $S^3$ and the involutive bordered Heegaard Floer theory of its complement determine each other. In particular, given two knots $K_1$ and $K_2$, we prove that the $\mathbb{F}_2[U,V]/(UV)$-coefficient involutive knot Floer homology of $K_1 \sharp -K_2$ is $ι_K$-locally trivial if $\widehat{CFD}(S^3 \backslash K_1)$ and $\widehat{CFD}(S^2 \backslash K_2)$ satisfy a certain condition which can be seen as the bordered counterpart of $ι_K$-local equivalence. We further establish an expli…
Using a Heegaard diagram for the pullback of a knot $K \subset S^3$ in its cyclic double branched cover $Σ_2(K)$, we give a combinatorial proof for the invariance of knot Floer homology over $\mathbb{Z}$.
Knot Floer homology is a knot invariant defined using holomorphic curves. In more recent work, taking cues from bordered Floer homology,the authors described another knot invariant, called "bordered knot Floer homology", which has an explicit algebraic and combinatorial construction. In the present paper, we extend the holomorphic theory to bordered Heegaard diagrams for partial knot projections, and establish a pairing result for gluing such diagrams, in the spirit of the pairing theorem of bordered Floer homology. After making some model calculations, we obtain an identification of a variant…
Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots.
The aim of this paper is to study the behavior of knot Floer homology under Murasugi sum. We establish a graded version of Ni's isomorphism between the extremal knot Floer homology of Murasugi sum of two links and the tensor product of the extremal knot Floer homology groups of the two summands. We further prove that $τ=g$ for each summand if and only if $τ=g$ holds for the Murasugi sum (with $τ$ and $g$ defined appropriately for multi-component links). Some applications are presented.
In an earlier work, we introduced a family of t-modified knot Floer homologies, defined by modifying the construction of knot Floer homology HFK-minus. The resulting groups were then used to define concordance homomorphisms indexed by t in [0,2]. In the present work we elaborate on the special case t=1, and call the corresponding modified knot Floer homology the unoriented knot Floer homology. Using elementary methods (based on grid diagrams and normal forms for surface cobordisms), we show that the resulting concordance homomorphism gives a lower bound for the smooth 4-dimensional crosscap nu…
We apply the theory of "peculiar modules" for the Floer homology of 4-ended tangles developed by Zibrowius (specifically, the immersed curve interpretation of the tangle invariants) to compute the Knot Floer Homology ($\widehat{HFK}$) of 3-stranded pretzel knots of the form ${P(2a,-2b-1,\pm(2c+1))}$ for positive integers $a,b,c$. This corrects a previous computation by Eftekhary; in particular, for the case of ${P(2a,-2b-1,2c+1)}$ where $b<c$ and $b<a-1$, it turns out the rank of $\widehat{HFK}$ is larger than that predicted by that work.
Ozsváth and Szabó conjectured that knot Floer homology detects fibred knots in $S^3$. We will prove this conjecture for null-homologous knots in arbitrary closed 3--manifolds. Namely, if $K$ is a knot in a closed 3--manifold $Y$, $Y-K$ is irreducible, and $\hat{HFK}(Y,K)$ is monic, then $K$ is fibred. The proof relies on previous works due to Gabai, Ozsváth--Szabó, Ghiggini and the author. A corollary is that if a knot in $S^3$ admits a lens space surgery, then the knot is fibred.
We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It carries information about the Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive information about other surgeries on knots, and about the maps on Floer homology induced by certain surgery cobordisms. We define a certain class of \em{perfect} knots in S^3 for which CF_r has a particularly simple form. For these knots, formal properties of the Ozsvath-Szabo theor…
The aim of this paper is to study the skein exact sequence for knot Floer homology. We prove precise graded version of this sequence, and also one using $\HFm$. Moreover, a complete argument is also given purely within the realm of grid diagrams.
We exhibit pairs of transverse knots with the same self-linking number that are not transversely isotopic, using the recently defined knot Floer homology invariant for transverse knots and some algebraic refinements of it.
This note corrects the mistakes in the splicing formulas of the paper "Floer homology and splicing knot complements". The mistakes are the result of the incorrect assumption that for a knot $K$ inside a homology sphere $Y$, the involution on the knot Floer homology of $K$ which corresponds to moving the basepoints by one full twist around $K$ is trivial. The correction implicitly involves considering the contribution from this (possibly non-trivial) involution in a number of places.
We show that if K is a non-trivial knot inside a homology sphere X, the rank of the knot Floer homology group associated with K is strictly bigger than the rank of the Heegaard Floer homology group associated with X.
We prove that knot Floer homology of a certain class of knots is non-trivial in next-to-top Alexander grading. This gives a partial affirmative answer to a question posed by Baldwin and Vela-Vick which asks if the same is true for all non-trivial knots in $S^3$.
In this note, we give a short proof that knot Floer thickness is a lower bound on the dealternating number of a knot. The result is originally due to work of Abe and Kishimoto, Lowrance, and Turaev. Our proof is a modification of the Stipsicz-Szabo approach using Kauffman states to show that thickness bounds the minimal number of bad domains in a knot diagram.
Knot Floer homology is an invariant for knots discovered by the authors and, independently, Jacob Rasmussen. The discovery of this invariant grew naturally out of studying how a certain three-manifold invariant, Heegaard Floer homology, changes as the three-manifold undergoes Dehn surgery along a knot. Since its original definition, thanks to the contributions of many researchers, knot Floer homology has emerged as a useful tool for studying knots in its own right. We give here a few selected highlights of this theory, and then move on to some new algebraic developments in the computation of k…
