Related papers: The knot Floer complex and the smooth concordance …
For pattern knots admitting genus-one bordered Heegaard diagrams, we show the knot Floer chain complexes of the corresponding satellite knots can be computed using immersed curves. This, in particular, gives a convenient way to compute the…
Using a Heegaard diagram for the pullback of a knot $K \subset S^3$ in its cyclic double branched cover $\Sigma_2(K)$, we give a combinatorial proof for the invariance of knot Floer homology over $\mathbb{Z}$.
We study three knot invariants related to smoothly immersed disks in the four-ball. These are the four-ball crossing number, which is the minimal number of normal double points of such a disk bounded by a given knot; the slicing number,…
Floer homology is a good example of homological invariants living in the infinite dimension. We suggest a way to construct this kind of invariants using only soft essentially finite-dimensional tools; no hard analysis or PDE is involved.…
Conjecturally, a knot in the 3-sphere has only finitely many non-integer non-characterizing slopes. We verify this conjecture for all knots with knot Floer homology satisfying certain simplicity conditions. The class of knots satisfying our…
The bipolar filtration of Cochran, Harvey and Horn presents a framework of the study of deeper structures in the smooth concordance group of topologically slice knots. We show that the graded quotient of the bipolar filtration of…
Given a connected cobordism between two knots in the 3-sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has…
We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and…
We develop and implement an algorithm that computes the full knot Floer complex of knots of thickness one. As an application, by extending this algorithm to certain knots of thickness two, we show that all but finitely many non-integral…
We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of…
We provide new information about the structure of the abelian group of topological concordance classes of knots in $S^3$. One consequence is that there is a subgroup of infinite rank consisting entirely of knots with vanishing Casson-Gordon…
In this paper we investigate the 0-concordance classes of 2-knots in $S^4$, an equivalence relation that is related to understanding smooth structures on 4-manifolds. Using Rochlin's invariant, and invariants arising from Heegaard-Floer…
We show that if the connected sum of two knots with coprime Alexander polynomials has vanishing von Neumann rho-invariants associated with certain metabelian representations then so do both knots. As an application, we give a new example of…
For any group G, we define a new characteristic series related to the derived series, that we call the torsion-free derived series of G. Using this series and the Cheeger-Gromov rho-invariant, we obtain new real-valued homology cobordism…
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…
To a nullhomologous knot $K$ in a 3-manifold $Y$, knot Floer homology associates a bigraded chain complex over $\mathbb{F}[U,V]$ as well as a collection of flip maps; we show that this data can be interpretted as a collection of decorated…
This thesis develops some general calculational techniques for finding the orders of knots in the topological concordance group C. The techniques currently available in the literature are either too theoretical, applying to only a small…
We discuss a concordance invariant constructed from Heegaard Floer homology "correction terms" and +/- 1 surgeries on knots in the three-sphere.
Using instanton Floer theory, extending methods due to Froyshov, we determine the definite lattices that arise from smooth 4-manifolds bounded by certain homology 3-spheres. For example, we show that for +1 surgery on the (2,5) torus knot,…
We use the contact invariant defined in [2] to construct a new invariant of Legendrian knots in Kronheimer and Mrowka's monopole knot homology theory (KHM), following a prescription of Stipsicz and V\'ertesi. Our Legendrian invariant…