arXiv subjects

Isabella Khan

Publications and source records attributed to Isabella Khan.

4 recordsLinked to original sources

arXiv2026

The Heegaard Floer d-invariant for more rational homology spheres

The Heegaard Floer d-invariant for a rational homology sphere Y and spin$^c$-structure $\mathfrak{s}$ is defined as the minimal absolute grading of a generator of $HF^+(Y; \mathfrak{s})$. In 2005, N\'emethi used lattice homology to compute the d-invariant for a particular class of negative-definite plumbed rational homology spheres, and conjectured that his formula should hold for all negative-definite plumbed rational homology spheres. In this paper, we use Zemke's isomorphism between lattice and Heegaard Floer homology to prove N\'emethi's conjecture.

math.GT↗

arXiv2025

Koszul dual $\mathcal{A}_{\infty}$-algebras from star-shaped diagrams -- part 2

This paper proves a Koszul duality result between weighted $\mathcal{A}_{\infty}$-algebras constructed in the author's previous work. In the process, we construct a new box tensor product for weighted $\mathcal{A}_{\infty}$ bimodules, and verify a correspondence between weighted $\mathcal{A}_{\infty}$-algebra maps and a particular class of $\mathcal{A}_{\infty}$-bimodule. This paper is part 2 of arXiv:2408.01564.

math.GT↗

arXiv2025

Bordered algebras and the wrapped Fukaya category

This paper establishes an isomorphism between endomorphism algebras from the wrapped Fukaya category of a type of punctured surface, and the class of A-infinity algebras related to bordered knot Floer homology, called star algebras, which the author first constructed in her previous work. By viewing the star algebras as A-infinity deformations of underlying associative algebras and making several calculations with Hochschild cohomology, we verify that the star algebras are unique with a given set of generators and basic A-infinity relations. We then make model calculations in order to establish that the endomorphism algebras have these generators and basic operations, so that the desired isomorphism follows.

math.GT↗

arXiv2024

Koszul dual $\mathcal{A}_{\infty}$ algebras for star-shaped diagrams -- Part 1

By slicing the Heegaard diagram for a given $3$-manifold in a particular way, it is possible to construct $\mathcal{A}_{\infty}$-bimodules, the tensor product of which retrieves the Heegaard Floer homology of the original 3-manifold. The first step in this is to construct algebras corresponding to the individual slices. Here, we use the graphical calculus for $\mathcal{A}_{\infty}$-structures introduced by Lipshitz, Ozsv\'ath, and Thurston, to construct Koszul dual weighted $\mathcal{A}_{\infty}$-algebras $\mathcal{A}$ and $\mathcal{B}$, and dualizing bimodules for a particular star-shaped class of slice. The duality result is then proved in the sequel.

math.GT↗