Related papers: On concordances in 3-manifolds

For a cyclic covering map $(\Sigma,K) \to (\Sigma',K')$ between two pairs of a 3-manifold and a knot each, we describe the fundamental group $\pi_1(\Sigma \setminus K)$ in terms of $\pi_1(\Sigma' \setminus K')$. As a consequence, we give an…

In this survey we summarize results regarding the Kauffman bracket, HOMFLYPT, Kauffman 2-variable and Dubrovnik skein modules, and the Alexander polynomial of links in lens spaces, which we represent as mixed link diagrams. These invariants…

We study a notion of strict pseudoconvexity in the context of topologically (often unsmoothably) embedded 3-manifolds in complex surfaces. Topologically pseudoconvex (TPC) 3-manifolds behave similarly to their smooth analogues, cutting out…

We propose a way to derive polynomial invariants of closed, orientable $3$-manifolds from Heegaard diagrams via cellularly embedded graphs. Given a Heegaard diagram of an irreducible $3$-manifold $M$, we associate a Heegaard graph $G\subset…

This paper's theme is the relation between several classical and well-known objects: triangle Fuchsian groups, quasi-homogeneous singularities of plane curves, torus knot complements in the 3-sphere. Torus knots are the only nontrivial…

In his 1957 paper, John Milnor introduced a collection of invariants for links in $S^3$ detecting higher-order linking phenomena by studying lower central quotients of link groups and comparing them to those of the unlink. These invariants,…

We show that the EH class and the LOSS invariant of Legendrian knots in contact 3-manifolds are functorial under regular Lagrangian concordances in Weinstein cobordisms. This gives computable obstructions to the existence of regular…

We generalize the Guth--Katz joints theorem from lines to varieties. A special case says that $N$ planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not…

We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S^3. The key role is played by the SL(2,Z) transformation, which generates a general torus knot from the unknot. Applying…

This elementary article introduces easy-to-manage invariants of genus one knots in homology 3-spheres. To prove their invariance, we investigate properties of an invariant of 3-dimensional genus two homology handlebodies called the…

A knot in the 3-sphere is called doubly slice if it is a slice of an unknotted 2-sphere in the 4-sphere. We give a bi-sequence of new obstructions for a knot being doubly slice. We construct it following the idea of Cochran-Orr-Teichner's…

The following is a long-standing open question: "If the zero-framed surgeries on two knots in the 3-sphere are integral homology cobordant, are the knots themselves concordant?" We show that an obvious rational version of this question has…

In knot concordance three genera arise naturally, g(K), g_4(K), and g_c(K): these are the classical genus, the 4-ball genus, and the concordance genus, defined to be the minimum genus among all knots concordant to K. Clearly 0 <= g_4(K) <=…

Algebraic knots are known to be iterated torus knots and to admit L-space surgeries. However, Hedden proved that there are iterated torus knots that admit L-space surgeries but are not algebraic. We present an infinite family of such…

Heegaard Floer homology and knot Floer homology are powerful invariants of 3-manifolds and links respectively. L-space knots are knots which admit Dehn surgeries to 3-manifolds with Heegaard Floer homology of minimal rank. In this paper we…

We use the knot homology of Khovanov and Lee to construct link concordance invariants generalizing the Rasmussen $s$-invariant of knots. The relevant invariant for a link is a filtration on a vector space of dimension $2^{|L|}$. The basic…

We construct an infinite family of hyperbolic, homologically thin knots that are not quasi-alternating. To establish the latter, we argue that the branched double-cover of each knot in the family does not bound a negative definite…

Cubic complexes appear in the theory of finite type invariants so often that one can ascribe them to basic notions of the theory. In this paper we begin the exposition of finite type invariants from the `cubic' point of view. Finite type…

Let $\widehat{\mathcal{C}}_{\mathbb{Z}}$ denote the group of knots in homology spheres that bound homology balls, modulo smooth concordance in homology cobordisms. Answering a question of Matsumoto, the second author previously showed that…

The guts of a knot is an invariant defined for the knot complement by Agol-Zhang. Nearly fibered knots, which are defined as knots whose Floer homology has dimension two in the top Alexander grading, were introduced by Baldwin-Sivek. In…