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Akram Alishahi

Publications and source records attributed to Akram Alishahi.

15 recordsLinked to original sources
arXiv2026

Bordered Floer homology and incompressible surfaces

We show that bordered Heegaard Floer homology detects incompressible surfaces and bordered-sutured Floer homology detects partly boundary parallel tangles and bridges, in natural ways. For example, there is a bimodule Lambda so that the tensor product of CFD(Y) and Lambda is Hom-orthogonal to CFD(Y) if and only if the boundary of Y admits an essential compressing disk. In the process, we sharpen a nonvanishing result of Ni's. We also extend Lipshitz-Ozsváth-Thurston's "factoring" algorithm for computing HF-hat to compute bordered-sutured Floer homology, to make both results on detecting incompressibility practical. In particular, this makes Zarev's tangle invariant manifestly combinatorial.

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arXiv2025

Colored knot Floer homology: structures and examples

Inspired by the $S^n$ colored version of Khovanov and Khovanov-Rozansky homology, we define a colored version of knot Floer homology by studying the colimit of a directed system of link Floer homology with infinite full twists. Specifically, our $n$-colored knot Floer homology of a knot $K$ is then defined as the colimit of the link Floer homology of $(n, mn)$-cables of $K$ by fixing $n$ and letting $m$ goes to infinity. We show that the colimit of the infinite full twists is a module over the colored knot Floer homology of the unknot. In addition, we give an explicit description of colored Heegaard Floer homology for L-space knots, and maps for colored knot Floer homology of crossing changes.

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arXiv2025

Detecting Heegaard Floer homology solid tori

We show that a rational homology solid torus is a Heegaard Floer homology solid torus if and only if it has a Dehn filling with a non-separating 2-sphere. Using this, we characterize Seifert fibered Heegaard Floer solid tori.

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arXiv2025

Bordered Floer homology, handlebody detection, and compressing diffeomorphisms

We show that, up to connected sums with integer homology $L$-spaces, bordered Floer homology detects handlebodies, as well as whether a mapping class extends over a given handlebody or compression body. Using this, we combine ideas of Casson-Long with the theory of train tracks to give an algorithm using bordered Floer homology to detect whether a mapping class extends over any compression body.

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arXiv2023

Splitting maps in link Floer homology and integer points in permutahedra

In this paper, we study the skein exact sequence for links via the exact surgery triangle of link Floer homology and compare it with other skein exact sequences given by Ozsváth and Szabó. As an application, we use the skein exact sequence to study the splitting number and splitting maps for links. In particular, we associate the splitting maps for the torus link $T(n, n)$ to integer points in the $(n-1)$-dimensional permutahedron, and obtain the link Floer homology of an $n$-component homology nontrivial unlink in $S^{1}\times S^{2}$.

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arXiv2023

Khovanov homology and the Involutive Heegaard Floer homology of branched double covers

We use involutive Heegaard Floer homology to extend the Ozsváth-Szabó branched double cover spectral sequence relating a version of Khovanov homology and the Heegaard Floer homology of branched double covers. Our main tools are Lipshitz, Ozsváth, and Thurston's reconstruction of the Ozsváth-Szabó spectral sequence using bordered Floer homology and Hendricks and Lipshitz's surgery exact triangle in involutive bordered Floer homology.

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arXiv2023

Bordered Floer homology and contact structures

We introduce a contact invariant in the bordered sutured Heegaard Floer homology of a three-manifold with boundary. The input for the invariant is a contact manifold $(M, ξ, \mathcal{F})$ whose convex boundary is equipped with a signed singular foliation $\mathcal{F}$ closely related to the characteristic foliation. Such a manifold admits a family of foliated open book decompositions classified by a Giroux Correspondence, as described in earlier work of Licata and Vértesi. We use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes $c_D$ and $c_A$ in the corresponding bordered sutured modules. Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. We also consider a natural map associated to forgetting the foliation $\mathcal{F}$ in favor of the dividing set, and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda-Kazez-Matić.

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arXiv2022

Upsilon invariant for graphs and the homology cobordism group of homology cylinders

Upsilon is a homomorphism on the smooth concordance group of knots defined by Ozsváth, Stipsicz and Szabó. In this paper, we define a generalization of upsilon for a family of embedded graphs in rational homolog spheres. We show that our invariant will induce a homomorphism on the homology cobordism group of homology cylinders, and present some applications. To define this invariant, we use tangle Floer homology. We lift relative gradings on tangle Floer homology to absolute gradings (for certain tangles) and prove a concatenation formula for it.

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arXiv2022

A friendly introduction to the bordered contact invariant

We give a short introduction to the contact invariant in bordered Floer homology defined by Földvári, Hendricks, and the authors. The construction relies on a special class of foliated open books. We discuss a procedure to obtain such a foliated open book and present a definition of the contact invariant. We also provide a "local proof", through an explicit bordered computation, of the vanishing of the contact invariant for overtwisted structures.

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arXiv2019

Relating tangle invariants for Khovanov homology and knot Floer homology

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 same algebras. For a closed diagram, this invariant computes the Khovanov homology of the knot or link. We show that forgetting the filtration, our DA bimodules are homotopy equivalent to a suitable version of the Ozsvath- Szabo bimodules. In addition to giving a relationship between tangle invariants for Khovanov homology and knot Floer homology, this gives an oriented skein exact triangle for the Ozsvath-Szabo bimodules which can be iterated to give an oriented cube of resolutions for the global construction.

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arXiv2018

A link invariant related to Khovanov homology and knot Floer homology

In this paper we introduce a chain complex $C_{1 \pm 1}(D)$ where D is a plat braid diagram for a knot K. This complex is inspired by knot Floer homology, but it the construction is purely algebraic. It is constructed as an oriented cube of resolutions with differential d=d_0+d_1. We show that the E_2 page of the associated spectral sequence is isomorphic to the Khovanov homology of K, and that the total homology is a link invariant which we conjecture is isomorphic to δ-graded knot Floer homology. The complex can be refined to a tangle invariant for braids on 2n strands, where the associated invariant is a bimodule over an algebra A_n. We show that A_n is isomorphic to B'(2n+1, n), the algebra used for the DA-bimodule constructed by Ozsvath and Szabo in their algebraic construction of knot Floer homology.

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arXiv2018

Tangle Floer homology and cobordisms between tangles

We introduce a generalization of oriented tangles, which are still called tangles, so that they are in one-to-one correspondence with the sutured manifolds. We define cobordisms between sutured manifolds (tangles) by generalizing cobordisms between oriented tangles. For every commutative algebra A over Z/2Z, we define A-Tangles to be the category consisting of A-tangles, which are balanced tangles with A-colorings of the tangle strands and fixed SpinC structures, and A-cobordisms as morphisms. An A-cobordism is a cobordism with a compatible A-coloring and an affine set of SpinC structures. Associated with every A-module M we construct a functor $HF^M$ from A-Tangles to A-Modules, called the tangle Floer homology functor, where A-Modules denotes the the category of A-modules and A-homomorphisms between them. Moreover, for any A-tangle T the A-module $HF^M(T)$ is the extension of sutured Floer homology defined in an earlier work of the authors. In particular, this construction generalizes the 4-manifold invariants of Ozsvath and Szabo. Moreover, applying the above machinery to decorated cobordisms between links, we get functorial maps on link Floer homology.

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arXiv2018

Knot Floer homology and the unknotting number

Given a knot K in S^3, let u^-(K) (respectively, u^+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants l^-(K), l^+(K) and l(K), which give lower bounds on u^-(K), u^+(K) and the unknotting number u(K), respectively. The invariant l(K) only vanishes for the unknot, and is greater than or equal to the ν^-(K). Moreover, the difference l(K)-ν^-(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance.

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arXiv2017

The Bar-Natan homology and unknotting number

We show that the order of torsion homology classes in Bar-Natan deformation of Khovanov homology is a lower bound for the unknotting number. We give examples of knots that this is a better lower bound than |s(K)/2|, where s(K) is the Rasmussen s invariant defined by the Bar-Natan spectral sequence.

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