# Bordered invariants in low-dimensional topology.

@inproceedings{Kotelskiy2018BorderedII, title={Bordered invariants in low-dimensional topology.}, author={Artem Kotelskiy}, year={2018} }

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 boundary. We first associate to the surface an algebra A1. Then to an immersed curve L inside the surface we associate an A∞ module M(L) over A1. Then we prove that Lagrangian Floer homology HF∗(L0, L1) is isomorphic to a suitable algebraic pairing of modules M(L0) and M(L1). We apply this theory to the…

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## One Citation

The Fukaya category of the pillowcase, traceless character varieties, and Khovanov Cohomology

- Mathematics
- 2018

For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We…

## References

SHOWING 1-10 OF 22 REFERENCES

Atiyah-Floer conjecture: A formulation, a
strategy of proof and generalizations

- MathematicsProceedings of Symposia in Pure Mathematics
- 2018

Around 1988, Floer introduced two important theories: instanton Floer homology as invariants of 3-manifolds and Lagrangian Floer homology as invariants of pairs of Lagrangians in symplectic…

A note on the knot Floer homology of fibered knots

- MathematicsAlgebraic & Geometric Topology
- 2018

We prove that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. Immediate applications include new proofs of Krcatovich's result that knots with $L$-space…

Bordered Floer homology for sutured manifolds

- Mathematics
- 2009

We define a sutured cobordism category of surfaces with boundary and 3-manifolds with corners. In this category a sutured 3-manifold is regarded as a morphism from the empty surface to itself. In the…

HF=HM, II : Reeb orbits and holomorphic
curves for the ech/Heegaard Floer correspondence

- MathematicsGeometry & Topology
- 2020

This is the second of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is…

Fukaya categories and bordered Heegaard-Floer homology

- Mathematics
- 2010

We outline an interpretation of Heegaard-Floer homology of 3-manifolds (closed or with boundary) in terms of the symplectic topology of symmetric products of Riemann surfaces, as suggested by recent…

Symplectic fixed points and holomorphic spheres

- Mathematics
- 1989

LetP be a symplectic manifold whose symplectic form, integrated over the spheres inP, is proportional to its first Chern class. On the loop space ofP, we consider the variational theory of the…

Khovanov homology is an unknot-detector

- Mathematics
- 2010

We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced…

Fukaya 𝐴_{∞}-structures associated to
Lefschetz fibrations. IV

- MathematicsProceedings of Symposia in Pure Mathematics
- 2019

We consider the Fukaya category associated to a basis of vanishing cycles in a Lefschetz fibration. We show that each element of the Floer cohomology of the monodromy around ∞ gives rise to a natural…

Combinatorial proofs in bordered Heegaard Floer homology

- Mathematics
- 2016

Using bordered Floer theory, we give a combinatorial construction and proof of invariance for the hat version of Heegaard Floer homology. As a part of the proof, we also establish combinatorially the…