# On combinatorial link Floer homology

@article{Manolescu2006OnCL, title={On combinatorial link Floer homology}, author={Ciprian Manolescu and Peter S. Ozsvath and Zolt{\'a}n Imre Szab{\'o} and Dylan P. Thurston}, journal={arXiv: Geometric Topology}, year={2006} }

Link Floer homology is an invariant for links defined using a suitable version of Lagrangian Floer homology. In an earlier paper, this invariant was given a combinatorial description with mod 2 coefficients. In the present paper, we give a self-contained presentation of the basic properties of link Floer homology, including an elementary proof of its invariance. We also fix signs for the differentials, so that the theory is defined with integer coefficients.

## Figures from this paper

figure 1 figure 2 figure 3 figure 4 figure 5 figure 6 figure 7 figure 8 figure 9 figure 10 figure 11 figure 12 figure 13 figure 14 figure 15 figure 16 figure 17 figure 18 figure 19 figure 20 figure 21 figure 22 figure 23 figure 24 figure 25 figure 26 figure 27 figure 28 figure 29 figure 30 figure 31 figure 32 figure 33 figure 34 figure 35

## 146 Citations

Sign refinement for combinatorial link Floer homology

- Mathematics
- 2008

Link Floer homology is an invariant for links which has recently been described entirely in a combinatorial way. Originally constructed with mod 2 coefficients, it was generalized to integer…

Singular link Floer homology

- Mathematics
- 2009

We define a grid presentation for singular links, ie links with a finite number of rigid transverse double points. Then we use it to generalize link Floer homology to singular links. Besides the…

Link Floer Homology Categorifies the Conway Function

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2016

Abstract Given an oriented link in the 3-sphere, the Euler characteristic of its link Floer homology is known to coincide with its multi-variable Alexander polynomial, an invariant only defined up to…

Transverse braids and combinatorial knot Floer homology

- Mathematics
- 2017

We describe a new method for combinatorially computing the transverse invariant in knot Floer homology. Previous work of the authors and Stone used braid diagrams to combinatorially compute knot…

A combinatorial spanning tree model for knot Floer homology

- Mathematics
- 2012

Abstract We iterate Manolescu’s unoriented skein exact triangle in knot Floer homology with coefficients in the field of rational functions over Z / 2 Z . The result is a spectral sequence which…

Combinatorial tangle Floer homology

- Mathematics
- 2014

In this paper we extend the idea of bordered Floer homology to knots and links in $S^3$: Using a specific Heegaard diagram, we construct gluable combinatorial invariants of tangles in $S^3$, $D^3$…

A spectral sequence on lattice homology

- Mathematics
- 2012

Using the link surgery formula for Heegaard Floer homology we find a spectral sequence from the lattice homology of a plumbing tree to the Heegaard Floer homology of the corresponding 3-manifold.…

Legendrian knots, transverse knots and combinatorial Floer homology

- Mathematics
- 2006

Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also…

On the Khovanov and knot Floer homologies of quasi-alternating links

- Mathematics
- 2007

Quasi-alternating links are a natural generalization of alternating links. In this paper, we show that quasi-alternating links are "homologically thin" for both Khovanov homology and knot Floer…

An introduction to tangle Floer homology

- Mathematics
- 2016

This paper is a short introduction to the combinatorial version of tangle Floer homology defined in "Combinatorial tangle Floer homology". There are two equivalent definitions---one in terms of…

## References

SHOWING 1-10 OF 29 REFERENCES

Holomorphic disks and knot invariants

- Mathematics
- 2002

Abstract We define a Floer-homology invariant for knots in an oriented three-manifold, closely related to the Heegaard Floer homologies for three-manifolds defined in an earlier paper. We set up…

A combinatorial description of knot Floer homology

- Mathematics
- 2006

Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are…

Khovanov homology and the slice genus

- Mathematics
- 2004

We use Lee’s work on the Khovanov homology to define a knot invariant s. We show that s(K) is a concordance invariant and that it provides a lower bound for the smooth slice genus of K. As a…

Knot Floer homology and the four-ball genus

- Mathematics
- 2003

We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance…

Floer homology and knot complements

- Mathematics
- 2003

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…

Holomorphic disks and topological invariants for closed three-manifolds

- Mathematics
- 2001

The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y, equipped with a Spiny structure. Given a Heegaard splitting of Y = U 0o U Σ U 1 , these…

Computations of Heegaard-Floer knot homology

- Mathematics
- 2006

We compute the knot Floer homology of knots with at most 12 crossings, as well as the τ invariant for knots with at most 11 crossings, using the combinatorial approach described by Manolescu, Ozsvath…

Embedding knots and links in an open book I: Basic properties

- Mathematics
- 1995

Abstract Birman and Menasco recently introduced a new way of presenting knots and links together with a corresponding link invariant. This paper examines the fundamental properties of this…

Holomorphic disks, link invariants and the multi-variable Alexander polynomial

- Mathematics
- 2008

We define a Floer-homology invariant for links in S 3 , and study its properties .

Morse theory for Lagrangian intersections

- Mathematics
- 1988

Soit P une variete symplectique compacte et soit L⊂P une sous-variete lagrangienne avec π 2 (P,L)=0. Pour un diffeomorphisme exact φ de P avec la propriete que φ(L) coupe L transversalement, on…