Related papers: Shake Slice and Shake Concordant Links
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 propose and analyze a structure with which to organize the difference between a knot in the 3-sphere bounding a topologically embedded 2-disk in the 4-ball and it bounding a smoothly embedded disk. The n-solvable filtration of the…
We present a new invariant, called slope, of a colored link in an integral homology sphere and use this invariant to complete the signature formula for the splice of two links. We develop a number of ways of computing the slope and a few…
Kawauchi proved that every strongly negative amphichiral knot $K \subset S^3$ bounds a smoothly embedded disk in some rational homology ball $V_K$, whose construction a priori depends on $K$. We show that $V_K$ is independent of $K$ up to…
We study the set $\widehat{\mathcal S}_M$ of framed smoothly slice links which lie on the boundary of the complement of a 1-handlebody in a closed, simply connected, smooth 4-manifold $M$. We show that $\widehat{\mathcal S}_M$ is…
An invariant of orientable 3-manifolds is defined by taking the minimum $n$ such that a given 3-manifold embeds in the connected sum of $n$ copies of $S^2 \times S^2$, and we call this $n$ the embedding number of the 3-manifold. We give…
We construct smooth manifolds with order two $\pi_1$ and even intersection forms which are irreducible, meaning they do not decompose into non-trivial connected sums. Their intersection forms being even implies that their universal covers…
We investigate the concordance properties of `parallel links' P(K), given by the (2,0) cable of a knot K. We focus on the question: if P(K) is concordant to a split link, is K necessarily slice? We show that if P(K) is smoothly concordant…
The bipolar filtration introduced by T. Cochran, S. Harvey, and P. Horn is a framework for the study of smooth concordance of topologically slice knots and links. It is known that there are topologically slice 1-bipolar knots which are not…
A knot in $S^3$ is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all…
A cobordism between links in thickened surfaces consists of a surface $ S $ and a $3$-manifold $M $, with $ S $ properly embedded in $ M \times I $. We show that there exist links in thickened surfaces such that if $(S,M) $ is a cobordism…
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 define a notion of complexity for shake-slice knots which is analogous to the definition of complexity for h-cobordisms studied by Morgan-Szab\'o. We prove that for each framing $n \ne 0$ and complexity $c \ge 0$, there is an…
The purpose of this paper is to study geometrically simply-connected homotopy 4-spheres by analyzing $n$-component links with a Dehn surgery realizing $\#^n(S^1\times S^2)$. We call such links $n$R-links. Our main result is that a homotopy…
We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the…
We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional…
We construct analogs of Khovanov-Jacobsson classes and the Rasmussen invariant for links in the boundary of any smooth oriented 4-manifold. The main tools are skein lasagna modules based on equivariant and deformed versions of…
By use of a variety of techniques (most based on constructions of quasipositive knots and links, some old and others new), many smooth 3-manifolds are realized as transverse intersections of complex surfaces in complex 3-space with strictly…
It is shown that if a link in 3-space bounds a proper oriented surface (without closed component) in the upper half 4-space, then the link bounds a proper oriented ribbon surface in the upper half 4-space which is a renewal embedding of the…
One strategy for distinguishing smooth structures on closed $4$-manifolds is to produce a knot $K$ in $S^3$ that is slice in one smooth filling $W$ of $S^3$ but not slice in some homeomorphic smooth filling $W'$. In this paper we explore…