Matthew Hedden

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 Floer homology of the cabled knot depends only on the filtered chain homotopy type of (K). A precise formula for this relationship is presented. In fact, the homology groups in the top 2 filtration dimensions for the cabled knot are isomorphic to the original knots Floer homology group in the top filtration dimension. The results are extended to (p,pn ±1) cables. As an example we compute HFK((T2,2m+1)2,2n+1) for all sufficiently large|n|, where T2,2m+1 denotes

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 HFK.D .K;t// in terms of the invariants for K , where D .K;t/ denotes the t ‐twisted positive (resp. negative)-clasped Whitehead double of K . In particular, the formula can be used iteratively and can be used to compute the Floer homology of manifolds obtained by surgery on Whitehead doubles. An immediate corollary is that . DC.K;t//D1 if t 0 are not smoothly slice. Another corollary is a closed formula for the Floer homology of the three-manifold obtained by gluing the complement of an arbitrary knot, K , to the complement of the trefoil.

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 generators, differentials, and rational Maslov and Alexander gradings in terms of combinatorial data on the grid diagram. Motivated by existing results for the Floer homology of knots in S 3 and the similarity of the resulting combinatorics presented here, we conjecture that a certain family of knots is characterized by their Floer homology. Coupled with the work of the third author, an affirmative answer to this would prove the Berge conjecture, which catalogs the knots in S 3 admitting lens space surgeries.

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 complete. The first step, which we prove here, is to show that a knot in a lens space with a threesphere surgery has simple (in the sense of rank) knot Floer homology. The second (conjectured) step involves showing that, for a fixed lens space, the only knots with simple Floer homology belong to a simple finite family. Using results of Baker, we provide evidence for the conjectural part of the program by showing that it holds for a certain family of knots. Coupled with work of Ni, these knots provide the first infinite family of non-trivial knots which are characterized by their knot Floer homology. As another application, we provide a Floer homology proof of a theorem of Berge.

The Ozsváth-Szabó and Rasmussen Concordance Invariants are not Equal

In this paper we present several counterexamples to Rasmussens conjecture that the concordance invariant coming from Khovanov homology is equal to twice the invariant coming from Ozsváth-Szabó Floer homology. The counterexamples are twisted Whitehead doubles of (2, 2n + 1) torus knots.

ON THE GEOGRAPHY AND BOTANY OF KNOT FLOER HOMOLOGY

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 detects the trefoil knot. For the second, we show that any non-trivial band sum of two unknots gives rise to an infinite family of distinct knots with isomorphic knot Floer homology. We also prove that the fibered knot with identity monodromy is strongly detected by its knot Floer homology, implying that Floer homology solves the word problem for mapping class groups of surfaces with non-empty boundary. Finally, we survey some conjectures and questions and, based on the results described above, formulate some new ones.

Knot concordance and homology cobordism

We consider the question: If the zero-framed surgeries on two oriented knots in the 3-sphere are integral homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant? We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on K is Z-homology cobordant to the zero-framed surgery on many of its winding number one satellites P(K). Then we prove that in many cases the tau and s-invariants of K and P(K) differ. Consequently neither tau nor s is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show, that a natural rational version of this question has a negative answer in both the topological and smooth categories, by proving similar results for K and its (p,1)-cables.

Khovanov module and the detection of unlinks

We study a module structure on Khovanov homology, which we show is natural under the Ozsvath–Szabo spectral sequence to the Floer homology of the branched double cover. As an application, we show that this module structure detects trivial links. A key ingredient of our proof is that the Λ∗H_1–module structure on Heegaard Floer homology detects S^1 × S^2 connected summands.

Does Khovanov homology detect the unknot

<abstract abstract-type=TeX><p> We determine a class of knots, which includes unknotting number one knots, within which Khovanov homology detects the unknot. A corollary is that the Khovanov homology of many satellite knots, including the Whitehead double, detects the unknot.

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 spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowkas singular instanton knot homology non-degenerate. The mechanism for this study is a (Lagrangian) intersection diagram which arises, through restriction of representations, from a tangle decomposition of a knot. When one of the tangles is trivial, our perturbations allow us to study isolated intersections of two Lagrangians to produce minimal generating sets for singular instanton knot homology. The (symplectic) manifold where this intersection occurs corresponds to the traceless character variety of the four-punctured 2-sphere, which we identify with the familiar pillowcase. We investigate the image in this pillowcase of the traceless representations of tangles obtained by removing a trivial tangle from 2-bridge knots and torus knots. Using this, we compute the singular instanton homology of a variety of torus knots. In many cases, our computations allow us to understand non-trivial differentials in the spectral sequence from Khovanov homology to singular instanton homology.