On the geography and botany of knot Floer homology

@article{Hedden2014OnTG,
  title={On the geography and botany of knot Floer homology},
  author={Matthew Hedden and Liam Watson},
  journal={Selecta Mathematica},
  year={2014},
  volume={24},
  pages={997-1037}
}
This paper explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology which do not arise as the invariant of a knot. This leads to a new constraint for knots admitting lens space surgeries, as well as a proof that the rank of knot Floer homology… 

Figures and Tables from this paper

A note on the knot Floer homology of fibered knots
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
Khovanov homology and knot Floer homology for pointed links
A well-known conjecture states that for any $l$-component link $L$ in $S^3$, the rank of the knot Floer homology of $L$ (over any field) is less than or equal to $2^{l-1}$ times the rank of the
Genus-two mutant knots with the same dimension in knot Floer and Khovanov homologies
We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology. Each knot in this infinite family admits a nontrivial genus two mutant which shares the same total dimension in
The cosmetic crossing conjecture for split links
Given a band sum of a split two-component link along a nontrivial band, we obtain a family of knots indexed by the integers by adding any number of full twists to the band. We show that the knots in
A note on a Geography problem in knot Floer homology
We prove that knot Floer homology of a certain class of knots is non-trivial in next-to-top Alexander grading. This gives a partial affirmative answer to a question posed by Baldwin and Vela-Vick
Knot Floer homology, link Floer homology and link detection.
We give new link detection results for knot and link Floer homology inspired by recent work on Khovanov homology. We show that knot Floer homology detects $T(2,4)$, $T(2,6)$, $T(3,3)$, $L7n1$, and
Links of Second Smallest Knot Floer Homology
We prove that the rank of knot Floer homology detects the Hopf links, and generalize this result further to classify the links of the second smallest knot Floer homology. We also prove a knot Floer
Heegaard Floer homology of Matsumoto's manifolds
We consider a homology sphere $M_n(K_1,K_2)$ presented by two knots $K_1,K_2$ with linking number 1 and framing $(0,n)$. We call the manifold {\it Matsumoto's manifold}. We show that there exists no
Khovanov homology detects the trefoils
We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the
A cobordism realizing crossing change on sl2 tangle homology and a categorified Vassiliev skein relation
Abstract We discuss degree-preserving crossing change on Khovanov homology in terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology, we investigate a sum of cobordisms that
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 110 REFERENCES
On Floer homology and the Berge conjecture on knots admitting lens space surgeries
We complete the first step in a two-part program proposed by Baker, Grigsby, and the author to prove that Berge’s construction of knots in the three-sphere which admit lens space surgeries is
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
On knot Floer homology and lens space surgeries
Abstract In an earlier paper, we used the absolute grading on Heegaard Floer homology HF + to give restrictions on knots in S 3 which admit lens space surgeries. The aim of the present article is to
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
On knot Floer homology and cabling
This paper is devoted to the study of the knot Floer homology groups \ HFK(S 3 ,K2,n), where K2,n denotes the (2,n) cable of an arbitrary knot, K. It is shown that for sufficiently large|n|, the
Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology
Similar to knots in S 3 , any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the
On Knot Floer Homology and Cabling: 2
We continue our study of the knot Floer homology invariants of cable knots. For large | n|, we prove that many of the filtered subcomplexes in the knot Floer homology filtration associated to the (p,
Knot Floer homology and the four-ball genus
We use the knot ltration on the Heegaard Floer complex d to dene an integer invariant (K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group
Knot Floer homology and the four-ball genus
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
Knot Floer homology of Whitehead doubles
In this paper we study the knot Floer homology invariants of the twisted and untwisted Whitehead doubles of an arbitrary knot, K . A formula is presented for the filtered chain homotopy type of b
...
1
2
3
4
5
...