Peculiar modules for 4‐ended tangles

@article{Zibrowius2020PeculiarMF,
  title={Peculiar modules for 4‐ended tangles},
  author={Claudius Zibrowius},
  journal={Journal of Topology},
  year={2020},
  volume={13}
}
With a 4‐ended tangle T , we associate a Heegaard Floer invariant CFT∂(T) , the peculiar module of T . Based on Zarev's bordered sutured Heegaard Floer theory (Zarev, PhD Thesis, Columbia University, 2011), we prove a glueing formula for this invariant which recovers link Floer homology HFL̂ . Moreover, we classify peculiar modules in terms of immersed curves on the 4‐punctured sphere. In fact, based on an algorithm of Hanselman, Rasmussen and Watson (Preprint, 2016, arXiv:1604.03466v2), we… 
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References

SHOWING 1-10 OF 64 REFERENCES
A cylindrical reformulation of Heegaard Floer homology
We reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold U a0;1c R, where U is the Heegaard surface, instead of Sym g .U/. We then show that the entire
Floer homology and knot complements
We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It
Bordered Heegaard Floer homology
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
Heegaard Floer homology and alternating knots.
In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y , which is closely related to the Heegaard Floer homology of Y . In this paper we
A calculus for bordered Floer homology
We consider a class of manifolds with torus boundary admitting bordered Heegaard Floer homology of a particularly simple form, namely, the type D structure may be described graphically by a disjoint
Joining and gluing sutured Floer homology
We give a partial characterization of bordered Floer homology in terms of sutured Floer homology. The bordered algebra and modules are direct sums of certain sutured Floer complexes. The algebra
The pillowcase and perturbations of traceless representations of knot groups
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
Bordered Floer homology for sutured manifolds
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
Tangle Floer homology and cobordisms between tangles
We introduce a generalization of oriented tangles, which are still called tangles, so that they are in one‐to‐one correspondence with the sutured manifolds. We define cobordisms between sutured
On a polynomial Alexander invariant for tangles and its categorification
We generalise the Kauffman state formula for the classical multivariate Alexander polynomial of knots and links to tangles and thereby obtain a finite set of polynomial tangle invariants. In the
...
...