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Roman Golovko

Publications and source records attributed to Roman Golovko.

At least 19 recordsLinked to original sources
arXiv2026

Non-decomposable Lagrangian endoconcordances and Khovanov homology

In this note, we construct non-decomposable Lagrangian endoconcordances of Legendrian knots. We prove that for any positive integer k, there exist a (stabilised) Legendrian knot and a Lagrangian fillable Legendrian knot in the standard contact 3-sphere, each admitting at least k pairwise Hamiltonian non-isotopic relative boundary, non-decomposable Lagrangian endoconcordances. The obstruction to decomposability that we rely on is based on Khovanov homology.

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arXiv2026

Non-regular Lagrangian concordances between Lagrangian fillable Legendrian knots

In this short note, we construct a family of non-regular, and therefore non-decomposable, Lagrangian concordances between Lagrangian fillable Legendrian knots in the standard contact 3-dimensional sphere. More precisely, for every decomposable Lagrangian concordance from $Λ_-$ to $Λ_+$, where $Λ_{\pm}$ are smoothly non-isotopic Legendrian knots, we construct a non-regular Lagrangian concordance from $Λ'_+$ to $Λ'_-$, where both $Λ'_\pm$ are Lagrangian fillable Legendrian knots obtained from $Λ_{\pm}$ by sufficiently many positive and negative stabilisations, followed by a sequence of Legendrian satellite operations.

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arXiv2026

Instability of Legendrian knottedness, and non-regular Lagrangian concordances of knots

We show that the family of smoothly non-isotopic Legendrian pretzel knots from the work of Cornwell-Ng-Sivek that all have the same Legendrian invariants as the standard unknot have front-spuns that are Legendrian isotopic to the front-spun of the unknot. Besides that, we construct the first examples of Lagrangian concordances between Legendrian knots that are not regular, and hence not decomposable. Finally, we show that the relation of Lagrangian concordance between Legendrian knots is not anti-symmetric, and hence does not define a partial order. The latter two results are based upon a new type of flexibility for Lagrangian concordances with stabilised Legendrian ends.

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arXiv2026

Non-decomposable Lagrangian cobordisms between Legendrian knots

For a given $g>0$, we construct a family of non-decomposable Lagrangian cobordisms of genus $g$ between (stabilized) Legendrian knots in the standard contact three-sphere. The main technique we use to obstruct decomposability is based on Livingston's estimates.

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arXiv2025

Lagrangian concordance is not a partial order in high dimensions

In this short note we provide the examples of pairs of closed, connected Legendrian non-isotopic Legendrian submanifolds $(Λ_{-}, Λ_{+})$ of the $(4n+1)$-dimensional contact vector space, $n>1$, such that there exist Lagrangian concordances from $Λ_-$ to $Λ_+$ and from $Λ_+$ to $Λ_-$. This contradicts anti-symmetry of the Lagrangian concordance relation, and, in particular, implies that Lagrangian concordances with connected Legendrian ends do not define a partial order in high dimensions. In addition, we explain how to get the same result for the relation given by exact Lagrangian cobordisms with connected Legendrian ends in the $(2n+1)$-dimensional contact vector space, $n>1$.

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arXiv2023

On non-geometric augmentations in high dimensions

In this note we construct augmentations of Chekanov-Eliashberg algebras of certain high dimensional Legendrian submanifolds that are not induced by exact Lagrangian fillings. The obstructions to the existence of exact Lagrangian fillings that we use are Seidel's isomorphism and the injectivity of a certain algebraic map between the corresponding augmentation varieties proven by Gao and Rutherford. In addition, along the way we discuss the relation between augmentation varieties of Legendrian submanifolds and their spherical spuns.

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arXiv2023

On torsion in linearized Legendrian contact homology

In this short note we discuss certain examples of Legendrian submanifolds, whose linearized Legendrian contact (co)homology groups over integers have non-vanishing algebraic torsion. More precisely, for a given arbitrary finitely generated abelian group $G$ and a positive integer $n\geq 3$, $n\neq 4$, we construct examples of Legendrian submanifolds of the standard contact vector space $\mathbb R^{2n+1}$, whose $n-1$-th linearized Legendrian contact (co)homology over $\mathbb Z$ computed with respect to a certain augmentation is isomorphic to $G$.

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arXiv2022

A note on infinite number of exact Lagrangian fillings for spherical spuns

In this short note we discuss high-dimensional examples of Legendrian submanifolds of the standard contact Euclidean space with an infinite number of exact Lagrangian fillings up to Hamiltonian isotopy. They are obtained from the examples of Casals and Ng by applying to them the spherical spinning construction.

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arXiv2022

Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors

We prove that the wrapped Fukaya category of any $2n$-dimensional Weinstein manifold (or, more generally, Weinstein sector) $W$ is generated by the unstable manifolds of the index $n$ critical points of its Liouville vector field. Our proof is geometric in nature, relying on a surgery formula for Floer cohomology and the fairly simple observation that Floer cohomology vanishes for Lagrangian submanifolds that can be disjoined from the isotropic skeleton of the Weinstein manifold. Note that we do not need any additional assumptions on this skeleton. By applying our generation result to the diagonal in the product $W \times W$, we obtain as a corollary that the open-closed map from the Hochschild homology of the wrapped Fukaya category of $W$ to its symplectic cohomology is an isomorphism, proving a conjecture of Seidel. We work mainly in the "linear setup" for the wrapped Fukaya category, but we also extend the proofs to the "quadratic" and "localisation" setup. This is necessary for dealing with Weinstein sectors and for the applications.

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arXiv2021

Legendrian submanifolds from Bohr-Sommerfeld covers of monotone Lagrangian tori

By a result due to Ziltener, there exist no closed embedded Bohr-Sommerfeld Lagrangians inside $\mathbb CP^n$ for the prequantisation bundle whose total space is the standard contact sphere. On the other hand, any embedded monotone Lagrangian torus has a canonical nontrivial cover which is a Bohr-Sommerfeld immersion. We draw the front projections for the corresponding Legendrian lifts inside a contact Darboux ball of the threefold covers of both the two-dimensional Clifford and Chekanov tori (the former is the Legendrian link of the Harvey-Lawson special Lagrangian cone), and compute the associated Chekanov-Eliashberg algebras. Although these Legendrians are not loose, we show that they both admit exact Lagrangian cobordisms to the loose Legendrian sphere; they hence admit exact Lagrangian caps in the symplectisation, which are non-regular Lagrangian cobordisms. Along the way, we also compute bilinearised Legendrian contact homology of a general Legendrian surface in the standard contact vector space when all Reeb chords are of positive degree, as well as the augmentation variety in the case of tori.

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arXiv2020

On Legendrian products and twist spuns

The Legendrian product of two Legendrian knots, as defined by Lambert-Cole, is a Legendrian torus. We show that this Legendrian torus is a twist spun whenever one of the Legendrian knot components is sufficiently large. We then study examples of Legendrian products which are not Legendrian isotopic to twist spuns. In order to do this, we prove a few structural results on the bilinearised Legendrian contact homology and augmentation variety of a twist spun. In addition, we show that the threefold Bohr-Sommerfeld covers of the Clifford torus and Chekanov torus are not twist spuns.

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arXiv2019

The stable Morse number as a lower bound for the number of Reeb chords

Assume that we are given a closed chord-generic Legendrian submanifold $Λ\subset P \times \mathbb R$ of the contactisation of a Liouville manifold, where $Λ$ moreover admits an exact Lagrangian filling $L_Λ \subset \mathbb R \times P \times \mathbb R$ inside the symplectisation. Under the further assumptions that this filling is spin and has vanishing Maslov class, we prove that the number of Reeb chords on $Λ$ is bounded from below by the stable Morse number of $L_Λ$. Given a general exact Lagrangian filling $L_Λ$, we show that the number of Reeb chords is bounded from below by a quantity depending on the homotopy type of $L_Λ$, following Ono-Pajitnov's implementation in Floer homology of invariants due to Sharko. This improves previously known bounds in terms of the Betti numbers of either $Λ$ or $L_Λ$.

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arXiv2018

Floer theory for Lagrangian cobordisms

In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the Chekanov-Eliashberg algebras of the negative ends of the cobordisms admit augmentations. From this theory we derive several exact sequences relating the Morse homology of an exact Lagrangian cobordism with the bilinearised contact homologies of its ends. These are then used to investigate the topological properties of exact Lagrangian cobordisms.

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arXiv2018

A Novikov fundamental group

Given a $1$-cohomology class $u$ on a closed manifold $M$, we define a Novikov fundamental group associated to $u$, generalizing the usual fundamental group in the same spirit as Novikov homology generalizes Morse homology to the case of non exact $1$-forms. As an application, lower bounds for the minimal number of index $1$ and $2$ critical points of Morse closed $1$-forms are obtained, that are different in nature from those derived from the Novikov homology.

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arXiv2016

Polyfolds: A First and Second Look

Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding commonalities in the analytic framework for a variety of geometric elliptic PDEs, in particular moduli spaces of pseudoholomorphic curves. It aims to systematically address the common difficulties of compactification and transversality with a new notion of smoothness on Banach spaces, new local models for differential geometry, and a nonlinear Fredholm theory in the new context. We shine meta-mathematical light on the bigger picture and core ideas of this theory. In addition, we compiled and condensed the core definitions and theorems of polyfold theory into a streamlined exposition, and outline their application at the example of Morse theory.

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arXiv2016

Noncommutative augmentation categories

To a differential graded algebra with coefficients in a noncommutative algebra, by dualisation we associate an $A_\infty$-category whose objects are augmentations. This generalises the augmentation category of Bourgeois and Chantraine to the noncommutative world.

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arXiv2016

Estimating the number of Reeb chords using a linear representation of the characteristic algebra

Given a chord-generic horizontally displaceable Legendrian submanifold $Λ\subset P\times \mathbb R$ with the property that its characteristic algebra admits a finite-dimensional matrix representation, we prove an Arnold-type lower bound for the number of Reeb chords on $Λ$. This result is a generalization of the results of Ekholm-Etnyre-Sullivan and Ekholm-Etnyre-Sabloff which hold for Legendrian submanifolds whose Chekanov-Eliashberg algebras admit augmentations. We also provide examples of Legendrian submanifolds $Λ$ of $\mathbb C^{n}\times \mathbb R$, $n \ge 1$, whose characteristic algebras admit finite-dimensional matrix representations, but whose Chekanov-Eliashberg algebras do not admit augmentations. In addition, to show the limits of the method of proof for the bound, we construct a Legendrian submanifold $Λ\subset \mathbb C^{n}\times \mathbb R$ with the property that the characteristic algebra of $Λ$ does not satisfy the rank property. Finally, in the case when a Legendrian submanifold $Λ$ has a non-acyclic Chekanov-Eliashberg algebra, using rather elementary algebraic techniques we obtain lower bounds for the number of Reeb chords of $Λ$. These bounds are slightly better than the number of Reeb chords that is possible to achieve with a Legendrian submanifold whose Chekanov-Eliashberg algebra is acyclic.

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