Related papers: Local Knots, $\nu^+$-Sharp Knots, and Rational Sli…

In an earlier work, we introduced a family of t-modified knot Floer homologies, defined by modifying the construction of knot Floer homology HFK-minus. The resulting groups were then used to define concordance homomorphisms indexed by t in…

We consider a homology sphere $M_n(K_1,K_2)$ presented by two knots $K_1,K_2$ with linking number 1 and framing $(0,n)$. We call the manifold {\it Matsumoto's manifold}. We show that there exists no contractible bound of $M_n(T_{2,3},K_2)$…

Ozsvath and Szabo proved that knot Floer homology determines the genera of knots in S^3. We will generalize this deep result to links in homology 3-spheres, by adapting their method. Our proof relies on a result of Gabai and some…

We determine the local equivalence class of the Seiberg-Witten Floer stable homotopy type of a spin rational homology 3-sphere $Y$ embedded into a spin rational homology $S^{1} \times S^{3}$ with a positive scalar curvature metric so that…

Ozsvath and Szabo defined an analog of the Froyshov invariant in the form of a correction term for the grading in Heegaard Floer homology. Applying this to the double cover of the 3-sphere branched over a knot K, we obtain an invariant…

A geometric argument is given to prove that the Seifert genus of a positive knot equals its slice genus. A combinatorial invariant, giving a lower bound for the slice genus, is formulated for arbitrary knots. Properties and applications of…

Using the conjugation symmetry on Heegaard Floer complexes, we define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to $\mathbb{Z}_4$-equivariant Seiberg-Witten Floer homology. Further,…

Extending Haken's Theorem to product annuli and disks for Heegaard splittings of sutured manifolds, we show that the handle number of an irreducible sutured manifold equals the handle number of its guts. We further show that reduced sutured…

The conormal Lagrangian $L_K$ of a knot $K$ in $\mathbb{R}^3$ is the submanifold of the cotangent bundle $T^* \mathbb{R}^3$ consisting of covectors along $K$ that annihilate tangent vectors to $K$. By intersecting with the unit cotangent…

The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman's theorem that Alexander polynomial one knots are topologically slice. This paper develops…

We discuss an obstruction to a knot being smoothly slice that comes from minimum-genus bounds on smoothly embedded surfaces in definite 4-manifolds. As an example, we provide an alternate proof of the fact that the (2,1)-cable of the figure…

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…

We compute the Heegaard Floer homology of $S^3_1(K)$ (the (+1) surgery on the torus knot $T_{p,q}$) in terms of the semigroup generated by $p$ and $q$, and we find a compact formula (involving Dedekind sums) for the corresponding…

We define a Rasmussen $s$-invariant over the coefficient ring of the integers, and show how it is related to the $s$-invariants defined over a field. A lower bound for the slice genus of a knot arising from it is obtained, and we give…

We prove that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. Immediate applications include new proofs of Krcatovich's result that knots with $L$-space surgeries are prime and Hedden and…

We formulate a refinement of SU(N) Chern-Simons theory on a three-manifold via the refined topological string and the (2,0) theory on N M5 branes. The refined Chern-Simons theory is defined on any three-manifold with a semi-free circle…

We describe an action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of almost-concordance between knots. After some basic results, we prove…

We show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance…

We prove that knot Floer homology of a certain class of knots is non-trivial in next-to-top Alexander grading. This gives a partial affirmative answer to a question posed by Baldwin and Vela-Vick which asks if the same is true for all…

There are several knot invariants in the literature that are defined using singular instantons. Such invariants provide strong tools to study the knot group and give topological applications. For instance, it gives powerful tools to study…