arXiv subjects

Melissa Zhang

Publications and source records attributed to Melissa Zhang.

13 recordsLinked to original sources

arXiv2026

Decompositions and diagrams of symplectic surfaces in Weinstein domains

We introduce combinatorial and diagrammatic methods for representing properly embedded symplectic surfaces in 4-dimensional Weinstein domains. We show that positive ascending surfaces, which include complex curves in Stein domains and multisections of Lefschetz fibrations, can be placed in bridge position with respect to Islambouli--Starkston's bisection-with-divides structure on the Weinstein domain. We algorithmically relate various decompositions of such surfaces, including transverse banded unlink diagrams, quasipositive factorizations, bridge bisections with divides, shadow diagrams (curves on surfaces), and pointed monodromy factorizations. We also develop a new way to present branched covers of Weinstein domains along positive ascending surfaces, which, combined with work of Loi--Piergallini, recovers Islambouli--Starkston's result that every compact Weinstein domain admits a bisection with divides.

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arXiv2026

6-valent vertex in the $\mathfrak{gl}_N$ web category and its categorification

We define a $2\pi/3$-rotationally invariant 6-valent vertex in the $\mathfrak{gl}_N$ web category. When $N = 4, 5$, we provide a categorification of the 6-valent vertex using $\mathfrak{gl}_N$ foams and decompose the hexagon web into a direct sum of indecomposables. A similar decomposition is conjectured for $N \geq 6$.

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arXiv2025

Khovanov skein lasagna modules with $1$-dimensional inputs

We construct a variant of Khovanov skein lasagna modules, which takes the Khovanov homology in connected sums of $S^1\times S^2$ defined by Rozansky and Willis as the input link homology. To carry out the construction, we prove functoriality of Rozansky-Willis's homology for cobordisms in a class of $4$-manifolds that we call $4$-dimensional relative $1$-handlebody complements, by using, as a bypass, an isomorphism proved by Sullivan--Zhang relating the Rozansky-Willis homology and the classical Khovanov skein lasagna module of links on the boundary of $D^2\times S^2$. Along the way, we also present new results on diffeomorphism groups, on Gluck twists for Khovanov skein lasagna modules, and on the functoriality of $\mathfrak{gl}_2$ foams.

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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.

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arXiv2025

Notes on Khovanov homology

These are expository lecture notes from a graduate topics course taught by the author on Khovanov homology and related invariants. Major topics include the Jones polynomial, Khovanov homology, Bar-Natan's cobordism category, applications of Khovanov homology, some spectral sequences, Khovanov stable homotopy type, and skein lasagna modules. Topological and algebraic exposition are sprinkled throughout as needed.

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arXiv2025

On the Categorified Wrapping Number Conjecture

We prove the Categorified Wrapping Number Conjecture for large classes of annular links, including alternating annular links and tangle closures exhibiting plumbed link phenomena. We do so by characterizing when a resolution is sufficient to produce a nonzero homology class in $k$-grading $\mathrm{wrap}(L)$ on its own. This characterization primarily concerns the type of crossing resolutions abutting trivial circles.

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arXiv2024

Kirby belts, categorified projectors, and the skein lasagna module of $S^{2}\times{S^{2}}$

We interpret Manolescu-Neithalath's cabled Khovanov homology formula for computing Morrison-Walker-Wedrich's $\mathrm{KhR}_2$ skein lasagna module as a homotopy colimit (mapping telescope) in a completion of the category of complexes over Bar-Natan's cobordism category. Using categorified projectors, we compute the $\mathrm{KhR}_2$ skein lasagna modules of (manifold, boundary link) pairs $(S^2 \times B^2, \tilde β)$, where $\tilde β$ is a geometrically essential boundary link, identifying a relationship between the lasagna module and the Rozansky projector appearing in the Rozansky-Willis invariant for nullhomologous links in $S^2 \times S^1$. As an application, we show that the $\mathrm{KhR}_2$ skein lasagna module of $S^2 \times S^2$ is trivial, confirming a conjecture of Manolescu.

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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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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.

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arXiv2022

Localization in Khovanov homology

We construct equivariant Khovanov spectra for periodic links, using the Burnside functor construction introduced by Lawson, Lipshitz, and Sarkar. By identifying the fixed-point sets, we obtain rank inequalities for odd and even Khovanov homologies, and their annular filtrations, for prime-periodic links in $S^3$.

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arXiv2020

On Khovanov Homology and Related Invariants

This paper begins with a survey of some applications of Khovanov homology to low-dimensional topology, with an eye toward extending these results to $\mathfrak{sl}(n)$ homologies. We extend Levine and Zemke's ribbon concordance obstruction from Khovanov homology to $\mathfrak{sl}(n)$ homology for $n \geq 2$, including the universal $\mathfrak{sl}(2)$ and $\mathfrak{sl}(3)$ homology theories. Inspired by Alishahi and Dowlin's bounds for the unknotting number coming from Khovanov homology and relying on spectral sequence arguments, we produce bounds on the alternation number of a knot. Lee and Bar-Natan spectral sequences also provide lower bounds on Turaev genus.

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arXiv2019

Annular link invariants from the Sarkar-Seed-Szabó spectral sequence

For a link in a thickened annulus $A \times I$, we define a $\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$ filtration on Sarkar-Seed-Szabó's perturbation of the geometric spectral sequence. The filtered chain homotopy type is an invariant of the isotopy class of the annular link. From this, we define a two-dimensional family of annular link invariants and study their behavior under cobordisms. In the case of annular links obtained from braid closures, we obtain a necessary condition for braid quasipositivity and a sufficient condition for right-veeringness, as well as Bennequin-type inequalities.

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arXiv2017

A rank inequality for the annular Khovanov homology of 2-periodic links

For a 2-periodic link $\tilde L$ in the thickened annulus and its quotient link $L$, we exhibit a spectral sequence with $E^1 \cong AKh(\tilde L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[θ, θ^{-1}] \rightrightarrows E^\infty \cong AKh(L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[θ, θ^{-1}].$ This spectral sequence splits along quantum and $sl_2$ weight space gradings, proving a rank inequality $rank\ AKh^{j,k}(L) \leq rank\ AKh^{2j-k,k} (\tilde L)$ for every pair of quantum and $sl_2$ weight space gradings $(j,k)$. We also present a few decategorified consequences and discuss partial results toward a similar statement for the Khovanov homology of 2-periodic links.

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