Related papers: Periodic knots and Heegaard Floer correction terms
Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse…
By proving a connected sum formula for the Legendrian invariant $\lambda_+$ in knot Floer homology we exhibit infinitely many transversely non simple knots.
We apply Heegaard Floer homology to study deformations of singularities of plane algebraic curves. Our main result provides an obstruction to the existence of a deformation between two singularities. Generalizations include the case of…
We review the use of grid diagrams in the development of Heegaard Floer theory. We describe the construction of the combinatorial link Floer complex, and the resulting algorithm for unknot detection. We also explain how grid diagrams can be…
By a recent result of Livingston, it is known that if a knot has a prime power branched cyclic cover that is not a homology sphere, then there is an infinite family of non-concordant knots having the same Seifert form as the knot. In this…
In this note we use Heegaard Floer homology to study smooth cobordisms of algebraic knots and complex deformations of cusp singularities of curves. The main tool will be the concordance invariant $\nu^+$: we study its behaviour with respect…
We introduce a framework for defining concordance invariants of knots using equivariant singular instanton Floer theory with Chern-Simons filtration. It is demonstrated that many of the concordance invariants defined using instantons in…
Let K \subset Y be a knot in a three manifold which admits a longitude-framed surgery such that the surgered manifold has first Betti number greater than that of Y. We find a formula which computes the twisted Floer homology of the surgered…
We introduce an unknotting-type number of knot projections that gives an upper bound of the crosscap number of knots. We determine the set of knot projections with the unknotting-type number at most two, and this result implies classical…
We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on…
Whitehead doubles provide a plethora of examples of knots that are topologically slice but not smoothly slice. We discuss the problem of the Whitehead double of the Figure 8 knot and survey commonly used techniques to obstructing sliceness.…
In this survey article, we discuss several different knot concordance invariants coming from the Heegaard Floer homology package of Ozsvath and Szabo. Along the way, we prove that if two knots are concordant, then their knot Floer complexes…
We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot.…
We study pseudo-classical knots in the non-orientable thickening of a non-orientable surface, specifically knots that are orientation-preserving paths in a non-orientable $3$-manifold of the form (non-orientable surface) $\times$ $[0, 1]$.…
We deal with Matveev complexity of compact orientable 3-manifolds represented via Heegaard diagrams. This lead us to the definition of modified Heegaard complexity of Heegaard diagrams and of manifolds. We define a class of manifolds which…
We address primary decomposition conjectures for knot concordance groups, which predict direct sum decompositions into primary parts. We show that the smooth concordance group of topologically slice knots has a large subgroup for which the…
In an earlier paper, we introduced a collection of graded Abelian groups $\HFKa(Y,K)$ associated to knots in a three-manifold. The aim of the present paper is to investigate these groups for several specific families of knots, including the…
The alternating knots, links and twists projected on the $S_2$ sphere were identified with the phase space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossings, the edges correspond…
We prove a formula for the involutive concordance invariants of the cabled knots in terms of that of the companion knot and the pattern knot. As a consequence, we show that any iterated cable of a knot with parameters of the form (odd,1) is…
We define a "reduced" version of the knot Floer complex $CFK^-(K)$, and show that it behaves well under connected sums and retains enough information to compute Heegaard Floer $d$-invariants of manifolds arising as surgeries on the knot…