Related papers: Upsilon type concordance invariants
We describe a new method for combinatorially computing the transverse invariant in knot Floer homology. Previous work of the authors and Stone used braid diagrams to combinatorially compute knot Floer homology of braid closures. However,…
Relative self-linking and linking "numbers" for pairs of knots in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual…
This manuscript introduces a new framework for the study of knots by exploring the neighborhood of knot embeddings in the space of simple open and closed curves in 3-space. The latter gives rise to a knotoid spectrum, which determines the…
In this paper we study embeddings of oriented connected closed surfaces in $\mathbb S^3$. We define a complete invariant, the fundamental span, for such embeddings, generalizing the notion of the peripheral system of a knot group. From the…
We extend knot contact homology to a theory over the ring $\mathbb{Z}[\lambda^{\pm 1},\mu^{\pm 1}]$, with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in $S^3$ and can be…
In this thesis we consider two constructions generalizing the classical Arf invariant. In the first construction an $\epsilon$-symmetric quadratic form over a ring with involution $R$ is lifted to an $\epsilon(1+T)$-symmetric quadratic form…
An oriented link L in a 3-sphere S in complex 2-space is a C-boundary if it bounds a piece of algebraic curve in the 4-ball bounded by S. Using Kronheimer and Mrowka's proof of the Thom Conjecture, we construct many oriented knots which are…
Inspired by the notions of local equivalence in monopole and Heegaard Floer homology, we introduce a version of local equivalence that combines odd Khovanov homology with equivariant even Khovanov homology into an algebraic package called a…
We relate the stability of knot invariants under twisting a pair of strands to the stability of symmetric quivers under unlinking (or linking) operation. Starting from the HOMFLY-PT skein relations, we confirm the stable growth of…
We recently defined an invariant of contact manifolds with convex boundary in Kronheimer and Mrowka's sutured monopole Floer homology theory. Here, we prove that there is an isomorphism between sutured monopole Floer homology and sutured…
Fixing two concordant links in $3$--space, we study the set of all embedded concordances between them, as knotted annuli in $4$--space. When regarded up to surface-concordance or link-homotopy, the set $\mathcal{C}(L)$ of concordances from…
We prove the equivalence of the invariants EH(L) and LOSS-(L) for oriented Legendrian knots L in the 3-sphere equipped with the standard contact structure, partially extending a previous result by Stipsicz and Vertesi. In the course of the…
It is well known that there are many 2-torsion elements in the classical knot concordance group. On the other hand, it is not known if there is any torsion element in the rational knot concordance group $\mathcal{C}_\mathbb{Q}$. Cha defined…
We establish a version of Seiberg--Witten Floer $K$-theory for knots, as well as a version of Seiberg-Witten Floer $K$-theory for 3-manifolds with involution. The main theorems are 10/8-type inequalities for knots and for involutions. The…
Various obstructions to knot concordance have been found using Casson-Gordon invariants, higher-order Alexander polynomials, as well as von-Neumann rho-invariants. Examples have been produced using (iterated) doubling operations K=R(c,J),…
Equivariant singular instanton Floer theory is a framework that associates to a knot in an integer homology 3-sphere a suite of homological invariants that are derived from circle-equivariant Morse-Floer theory of a Chern-Simons functional…
In this note, we describe a connection between the enumerative geometry of curves in K3 surfaces and the chiral ring of an auxiliary superconformal field theory. We consider the invariants calculated by Yau--Zaslow (capturing the Euler…
We describe an invariant of a contact 3-manifold with convex boundary as an element of Juh\'asz's sutured Floer homology. Our invariant generalizes the contact invariant in Heegaard Floer homology in the closed case, due to Ozsv\'ath and…
Ozsvath and Szabo have defined a knot concordance invariant tau that bounds the 4-ball genus of a knot. Here we discuss shortcuts to its computation. We include examples of Alexander polynomial one knots for which the invariant is…
The paper introduces Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the…