arXiv subjects
Publications and source records attributed to Matthew Hedden.
We determine the relationship between the contact structure induced by a fibered knot, K, in the three-sphere and the contact structures induced by its various cables. Understanding this relationship allows us to classify fibered cable knots which bound a properly embedded complex curve in the four-ball satisfying a genus constraint. This generalizes the well-known classification of links of plane curve singularities.
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 three-sphere 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.
Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsvath-Szabo contact invariant we obtain an invariant of knots in a contact three-manifold. This invariant provides an upper bound for the Thurston-Bennequin plus rotation number of any Legendrian realization of the knot. We use it to demonstrate the first systematic construction of prime knots in contact manifolds other than the three-sphere with negative maximal Thurston-Bennequin invariant. Perhaps more interesting, our invariant provides a criterion for an open book to induce a tight contact structure. A corollary is that if a manifold possesses contact structures with distinct non-vanishing Ozsvath-Szabo invariants, then any fibered knot can realize the classical Eliashberg-Bennequin bound in at most one of these contact structures.
In this paper we study the knot Floer homology invariants of the twisted and untwisted Whitehead doubles of an arbitrary knot K. We present a formula for the filtered chain homotopy type of HFK(D(+,K,t)) in terms of the invariants for K, where D(+,K,t) denotes the t-twisted positive-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 tau(D(+,K,t))=1 if t< 2tau(K) and zero otherwise, where tau is the Ozsv{á}th-Szab{ó} concordance invariant. It follows that the iterated untwisted Whitehead doubles of a knot satisfying tau(K)>0 are not smoothly slice.
In this paper we present several counterexamples to Rasmussen's conjecture that the concordance invariant coming from Khovanov homology is equal to twice the invariant coming from Ozsv{á}th-Szab{ó} Floer homology. The counterexamples are twisted Whitehead doubles of the (2,2n+1) torus knots.
This paper is devoted to the study of the knot Floer homology groups HFK(S^3,K_{2,n}), where K_{2,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 CFK(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 knot's Floer homology group in the top filtration dimension. The results are extended to (p,pn+-1) cables. As an example we compute HFK((T_{2,2m+1})_{2,2n+1}) for all sufficiently large |n|, where T_{2,2m+1} denotes the (2,2m+1)-torus knot.
In this paper we examine the relationship between various types of positivity for knots and the concodance invariant tau discovered by Ozsvath and Szabo and independently by Rasmussen. The main result shows that, for fibered knots, tau characterizes strong quasipositivity. This is quantified by the statement that for K fibered, tau(K)=g(K) if and only if K is strongly quasipositive. In addition, we survey existing results regarding tau and forms of positivity and highlight several consequences concerning the types of knots which are (strongly) (quasi) positive. For instance, we show that any knot known to admit a lens space surgery is strongly quasipositive and exhibit infinite families of knots which are not quasipositive.