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Rostislav Akhmechet

Publications and source records attributed to Rostislav Akhmechet.

11 recordsLinked to original sources
arXiv2025

Quantum topological Hochschild homology and annular Khovanov spectra

Topological Hochschild homology is a topological analogue of classical Hochschild homology of algebras and bimodules. Beliakova, Putyra, and Wehrli introduced quantum Hochschild homology (qHH) and used it to define a quantization of annular Khovanov homology as qHH of the tangle bimodules of Chen-Khovanov and Stroppel. After introducing quantum topological Hochschild homology (qTHH), we construct a new stable homotopy refinement of quantum annular Khovanov homology and show that it agrees with qTHH of the spectral Chen-Khovanov tangle bimodules of Lawson, Lipshitz, and Sarkar. We also show that this new stable homotopy refinement recovers the construction introduced in earlier work of Krushkal together with the first and third authors.

math.GT↗
arXiv2025

Annular SL(2) and SL(3) web algebras

We use annular foam TQFTs introduced by the first two authors to define equivariant $SL(2)$ and $SL(3)$ web algebras in the annulus. To a diagram of a tangle in the thickened annulus we assign a complex of bimodules over these algebras whose chain homotopy type is an invariant of the tangle. Several properties of algebras and bimodules are established. An essential technical part of the paper provides a bijective correspondence between non-elliptic annular $SL(3)$ webs and closed paths in the $SL(3)$ weight lattice. This generalizes an analogous bijection in the planar setting.

math.GT↗
arXiv2024

Khovanov homology and quantum error-correcting codes

Error-correcting codes for quantum computing are crucial to address the fundamental problem of communication in the presence of noise and imperfections. Audoux used Khovanov homology to define families of quantum error-correcting codes with desirable properties. We explore Khovanov homology and some of its many extensions, namely reduced, annular, and $\mathfrak{sl}_3$ homology, to generate new families of quantum codes and to establish several properties about codes that arise in this way, such as behavior of distance under Reidemeister moves or connected sums.

cs.IT↗
arXiv2024

Knot lattice homology and $q$-series invariants for plumbed knot complements

We introduce an invariant of negative definite plumbed knot complements unifying knot lattice homology, due to Ozsv\'ath, Stipsicz, and Szab\'o, and the BPS $q$-series of Gukov and Manolescu. This invariant is a natural extension of weighted graded roots of negative definite plumbed 3-manifolds introduced earlier by the first two authors and Krushkal. We prove a surgery formula relating our invariant with the weighted graded root of the surgered 3-manifold.

math.GT↗
arXiv2023

Lattice cohomology and $q$-series invariants of $3$-manifolds

An invariant is introduced for negative definite plumbed $3$-manifolds equipped with a spin$^c$-structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the study of normal surface singularities, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization gives BPS $q$-series which satisfy some remarkable modularity properties and recover ${\rm SU}(2)$ quantum invariants of $3$-manifolds at roots of unity. In particular, our work gives rise to a $2$-variable refinement of the $\widehat Z$-invariant.

math.GT↗
arXiv2022

Concordance invariants from $U(1) \times U(1)$-equivariant Khovanov homology

We study Khovanov homology over the Frobenius algebra $\mathbb{F}[U,V,X]/((X-U)(X-V))$, or $U(1) \times U(1)$-equivariant Khovanov homology, and extract two families of concordance invariants using the algebraic $U$-power and $V$-power filtrations on the chain complex. We also further develop the reduced version of the theory and study its behavior under mirroring.

math.GT↗
arXiv2021

Anchored foams and annular homology

We describe equivariant SL(2) and SL(3) homology for links in the solid torus via foam evaluation. The solid torus is replaced by 3-space with a distinguished line in it. Generators of state spaces for annular webs are represented by foams with boundary that may intersect the distinguished line; intersection points, called anchor points, contribute additional terms, reminiscent of square roots of the Hessian, to the foam evaluation. Both oriented and unoriented SL(3) foams are treated in the paper.

math.GT↗
arXiv2020

Towards an $\mathfrak{sl}_2$ action on the annular Khovanov spectrum

Given a link in the thickened annulus, its annular Khovanov homology carries an action of the Lie algebra $\mathfrak{sl}_2$, which is natural with respect to annular link cobordisms. We consider the problem of lifting this action to the stable homotopy refinement of the annular homology. As part of this program, the actions of the standard generators of $\mathfrak{sl}_2$ are lifted to maps of spectra. In particular, it follows that the $\mathfrak{sl}_2$ action on homology commutes with the action of the Steenrod algebra. The main new technical ingredients developed in this paper, which may be of independent interest, concern certain types of cancellations in the cube of resolutions and the resulting more intricate structure of the moduli spaces in the framed flow category.

math.GT↗
arXiv2020

Equivariant annular Khovanov homology

We construct an equivariant version of annular Khovanov homology via the Frobenius algebra associated with $U(1) \times U(1)$-equivariant cohomology of $\mathbb{CP}^1$. Motivated by the relationship between the Temperley-Lieb algebra and annular Khovanov homology, we also introduce an equivariant analogue of the Temperley-Lieb algebra.

math.QA↗
arXiv2019

Stable homotopy refinement of quantum annular homology

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an annular link $L$ a naive $\mathbb{Z}/r\mathbb{Z}$-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb{Z}[\mathbb{Z}/r\mathbb{Z}]$. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.

math.GT↗