arXiv research

Proves a new skein exact triangle for real monopole Floer homology.

New skein exact triangles for link Floer homology.

Defines band maps in unoriented link Floer homology forming a skein exact triangle.

We re-derive Manolescu's unoriented skein exact triangle for knot Floer homology over F_2 combinatorially using grid diagrams, and extend it to the case with Z coefficients by sign refinements. Iteration of the triangle gives a cube of resolutions that converges to the knot Floer homology of an oriented link. Finally, …

This paper defines a spectral sequence connecting knot homologies.

This work extends knot homology theory to links, proving exact triangles and categorifying link signatures.

The paper studies splitting maps in link Floer homology using skein exact sequences.

Given a crossing in a planar diagram of a link in the three-sphere, we show that the knot Floer homologies of the link and its two resolutions at that crossing are related by an exact triangle. As a consequence, we deduce that for any quasi-alternating knot, the total rank of its knot Floer homology is equal to the det…

We show that the bordered-sutured Floer invariant of the complement of a tangle in an arbitrary 3-manifold $Y$, with minimal conditions on the bordered-sutured structure, satisfies an unoriented skein exact triangle. This generalizes a theorem by Manolescu for links in $S^3$. We give a theoretical proof of this result …

New spectral sequence connects Khovanov homology to real monopole Floer homology.

The SO(3) instanton homology recently introduced by the authors associates a finite-dimensional vector space over the field of two elements to every embedded trivalent graph (or "web"). The present paper establishes a skein exact triangle for this instanton homology, as well as a realization of the octahedral axiom. Fr…

SU(3) instanton homology counts Tait colorings for webs and foams.

We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot…

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 $\widetilde{\mathrm{CT}} (T)$. If $L$ is obtained by gluing together $T_1, \dotsc, T_m$, then the knot Floer homology $\hat{\mathrm{HFK}}(L)…

By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface. The new skein algebra of an ideal triangle has a simple presentation. This gives …

Traditionally introduced in terms of advanced topological constructions, many link invariants may also be defined in much simpler terms given their values on a few initial links and a recursive formula on a skein triangle. Then the crucial question to ask is how many initial values are necessary to completely determine…

We iterate Manolescu's unoriented skein exact triangle in knot Floer homology with coefficients in the field of rational functions over $\mathbb{Z}/2\mathbb{Z}$. The result is a spectral sequence which converges to a stabilized version of delta-graded knot Floer homology. The $(E_2,d_2)$ page of this spectral sequence …

New surgery exact triangles in Heegaard Floer homology for rational slopes.

In this paper, we explore the interplay between contact structures and sutured monopole Floer homology. First, we study the behavior of contact elements, which were defined by Baldwin and Sivek, under the operation of performing Floer excisions, which was introduced to the context of sutured monopole Floer homology by …

Decomposes $SL_3$ skein algebras for surfaces.

Grid homology theory for spatial graphs extends skein sequence.

Study exact surgery formula in involutive Heegaard Floer homology.

This is the third part of the work on the exact triangles. We construct chain homomorphisms and show exactness of the resulting sequence.

Given an involution on a rational homology 3-sphere $Y$ with quotient the $3$-sphere, we prove a formula for the Lefschetz number of the map induced by this involution in the reduced monopole Floer homology. This formula is motivated by a variant of Witten's conjecture relating the Donaldson and Seiberg--Witten invaria…

Proves exact triangle linking knot instanton Floer homology to surgeries.

Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and there is a Poisson algebra homomorphism between the curve algebra an…

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.

In this paper, we consider the exact triangles consisting of stable vector bundles on one-dimensional complex tori, and give a geometric interpretation of them in terms of the corresponding Fukaya category via the homological mirror symmetry.

We prove the existence of an exact triangle for the Pin(2)-monopole Floer homology groups of three manifolds related by specific Dehn surgeries on a given knot. Unlike the counterpart in usual monopole Floer homology, only two of the three maps are those induced by the corresponding elementary cobordism. We use this tr…

Proves surgery exact triangle for monopole Floer homology over integers.

Ozsvath and Szabo recently constructed an algebraically defined invariant of tangles which takes the form of a DA bimodule. This invariant is expected to compute knot Floer homology. The authors have a similar construction for open braids and their plat closures which can be viewed as a filtered DA bimodule over the sa…

Combinatorial proof of grid homology properties.

We define a family of representations $\{ρ_n\}_{n\geq 0}$ of a pure braid group $P_{2k}$. These representations are obtained from an action of $P_{2k}$ on a certain type of $A_2$ web space with color $n$. The $A_2$ web space is a generalization of the Kauffman bracket skein module of a disk with marked points on its bo…

Let $(X^n, \check{X}^n)$ be a mirror pair of an $n$-dimensional complex torus $X^n$ and its mirror partner $\check{X}^n$. Then, a simple projectively flat bundle $E(L,\mathcal{L})\rightarrow X^n$ is constructed from each affine Lagrangian submanifold $L$ in $\check{X}^n$ with a unitary local system $\mathcal{L} \righta…

Paper computes a specific term of knot homology for 3-braids.

We study the algebraic and geometric properties of stated skein algebras of surfaces with punctured boundary. We prove that the skein algebra of the bigon is isomorphic to the quantum group ${\mathcal O}_{q^2}(\mathrm{SL}(2))$ providing a topological interpretation for its structure morphisms. We also show that its sta…

Proves properties of instanton knot Floer homology and connected sum formula.

New bounds on inscribed triangles in arbitrary planar domains.

We establish the exact triangle in Seiberg-Witten-Floer theory relating the monopoloe homologies of any two closed 3-manifolds which are obtained from each other by $\pm 1$-surgery. We also show that the sum of the modified version of the Seiberg-Witten invariants for any closed rational homology 3-sphere $Y$ over all …

Introduces integer-valued Heegaard Floer theory with canonical orientations.

We generalize our previous work on categorification of Kauffman bracket skein module of surfaces, by extending our homology to tangles in cylinders over surfaces, F x [0,1]. Our homology of 0-tangles and 1-tangles in D^3 coincides (up to normalization) with Khovanov link homology and the reduced Khovanov link homology.…

Lattice cohomology, defined by Némethi in (arXiv:0709.0841), is an invariant of negative definite plumbed 3-manifolds which conjecturally computes the Heegaard Floer homology HF^+. We prove a surgery exact triangle for the lattice cohomology analogous to the one for HF^+. This is a step towards comparing these two inva…

We construct a spectral sequence from the reduced odd Khovanov homology of a link converging to the framed instanton homology of the double cover branched over the link, with orientation reversed. Framed instanton homology counts certain instantons on the cylinder of a 3-manifold connect-summed with a 3-torus. En route…

Developed a theory of stated SL(n)-skein modules for 3-manifolds.

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 $SU(N)$-ins…

New CR representations are found and shown to be redundant.

This is the second part of the proof of the exact traiangles in Seiberg-Witten Floer theory. We analyse the splitting and gluing of flow lines of the Chern-Simons-Dirac functional when the underlying three-manifold splits along a torus. (two corrections added)

With a 4-ended tangle $T$, we associate a Heegaard Floer invariant $\operatorname{CFT^\partial}(T)$, the peculiar module of $T$. Based on Zarev's bordered sutured Heegaard Floer theory, we prove a glueing formula for this invariant which recovers link Floer homology $\operatorname{\widehat{HFL}}$. Moreover, we classify…