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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 bExpand
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|, theExpand
In this paper we examine the relationship between various types of positivity for knots and the concordance invariant τ discovered by Ozsvath and Szabo and independently by Rasmussen. The main resultExpand
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 theExpand
Topologically slice knots with nontrivial Alexander polynomial
Let C_T be the subgroup of the smooth knot concordance group generated by topologically slice knots and let C_D be the subgroup generated by knots with trivial Alexander polynomial. We prove theExpand
Splicing knot complements and bordered Floer homology
We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology rank strictly greater than one. InExpand
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?Expand
Non-slice linear combinations of algebraic knots
We show that the subgroup of the knot concordance group generated by links of isolated complex singularities intersects the subgroup of algebraically slice knots in an infinite rank subgroup.
On the functoriality of Khovanov–Floer theories
We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory assigns a filtered chain complex over Z/2 to a link diagram such that (1) the E_2 page of the resulting spectral sequence isExpand
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 isExpand