# Bordered theory for pillowcase homology

@article{Kotelskiy2019BorderedTF, title={Bordered theory for pillowcase homology}, author={Artem Kotelskiy}, journal={Mathematical Research Letters}, year={2019} }

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 pillowcase we associate an A infinity module M(L) over A. Then we prove that Lagrangian Floer homology HF(L,L') is isomorphic to a suitable algebraic pairing of modules M(L) and M(L'). This extends the pillowcase homology construction - given a 2-stranded tangle inside a 3-ball, if one obtains an…

## Figures from this paper

## 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 43 REFERENCES

The pillowcase and traceless representations of knot groups II: a Lagrangian-Floer theory in the pillowcase

- Mathematics
- 2014

We define an elementary relatively $\mathbb Z/4$ graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram…

Bordered Heegaard Floer homology

- MathematicsMemoirs of the American Mathematical Society
- 2018

We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with…

On bordered theories for Khovanov homology

- Mathematics
- 2015

We describe how to formulate Khovanov's functor-valued invariant of tangles in the language of bordered Heegaard Floer homology. We then give an alternate construction of Lawrence Roberts' Type D and…

The pillowcase and perturbations of traceless representations of knot groups

- Mathematics
- 2013

We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli…

Fukaya categories of symmetric products and bordered Heegaard-Floer homology

- Mathematics
- 2010

The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of Fukaya categories of symmetric products and…

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…

Bimodules in bordered Heegaard Floer homology

- Mathematics
- 2010

Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface F an algebra A(F) and to a three-manifold Y with boundary identified with F a module over A(F). In this…

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…