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Minimal and Hamiltonian-minimal submanifolds in toric geometry
In this paper we investigate a family of Hamiltonian-minimal Lagrangian submanifolds in ${\mathbb C}^m$, ${\mathbb C}P^m$ and other symplectic toric manifolds constructed from intersections of real
Immersed curves in Khovanov homology
We give a geometric interpretation of Bar-Natan's universal invariant for the class of tangles in the 3-ball with four ends: we associate with such 4-ended tangles $T$ multicurves
Khovanov invariants via Fukaya categories: the tangle invariants agree
Given a pointed 4-ended tangle $T \subset D^3$, there are two Khovanov theoretic tangle invariants, $\unicode{1044}_1(T)$ from [arXiv:1910.1458] and $L_T$ from [arXiv:1808.06957], which are twisted
Khovanov multicurves are linear
In previous work we introduced a Khovanov multicurve invariant K̃h associated with Conway tangles. Applying ideas from homological mirror symmetry we show that K̃h is subject to strong geography
There is a common main idea in various constructions of low-dimensional topological invariants. One takes a topological object, associates a geometric construction to it, which involves some choices,
Bordered theory for pillowcase homology
We construct an algebraic version of Lagrangian Floer homology for immersed curves inside the pillowcase. We first associate to the pillowcase an algebra A. Then to an immersed curve L inside the
Bordered invariants in low-dimensional topology.
In this thesis we present two projects. In the first project, which covers Chapters 2 and 3, we construct an algebraic version of Lagrangian Floer homology for immersed curves in a surface with
A mnemonic for the Lipshitz-Ozsv\'ath-Thurston correspondence.
When $\mathbf{k}$ is a field, type D structures over the algebra $\mathbf{k}[u,v]/(uv)$ are equivalent to immersed curves decorated with local systems in the twice-punctured disk. Consequently, knot
Cosmetic operations and Khovanov multicurves
We prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots. Our proof combines a recent result of Hanselman with the Khovanov multicurve invariants K̃h and B̃N.