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… 
View on Wiley
arxiv.org
15 Citations
Highly Influential Citations
1
Background Citations
7
Methods Citations
9
Results Citations
3

15 Citations

Citation Type

Citation Type

Heegaard Floer homology for manifolds with torus boundary: properties and examples
This is a companion paper to earlier work of the authors, which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We prove a…
ON SYMMETRIES OF PECULIAR MODULES or δ-GRADED LINK FLOER HOMOLOGY IS MUTATION INVARIANT
We investigate symmetry properties of peculiar modules, a Heegaard Floer invariant of 4-ended tangles which the author introduced in [Zib20]. In particular, we give an almost complete answer to the…
The Fukaya category of the pillowcase, traceless character varieties, and Khovanov Cohomology
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…
On symmetries of peculiar modules; or, $\delta$-graded link Floer homology is mutation invariant
We investigate symmetry properties of peculiar modules, a Heegaard Floer invariant of 4-ended tangles which the author introduced in [arXiv:1712.05050]. In particular, we give an almost complete…
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…
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…
The correspondence induced on the pillowcase by the earring tangle
The earring tangle consists of four strands 4pt x [0,1] inside S^2 x [0,1] and one meridian around one of the strands. Equipping this tangle with a nontrivial SO(3) bundle, we show that its traceless…
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…
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…
Kauffman states and Heegaard diagrams for tangles
We define polynomial tangle invariants $\nabla_T^s$ via Kauffman states and Alexander codes and investigate some of their properties. In particular, we prove symmetry relations for $\nabla_T^s$ of…
...
1
2
...

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…
...
1
2
3
4
5
...