Related papers: On Shake Slice Knots

Hyperfinite knots, or limits of equivalence classes of knots induced by a knot invariant taking values in a metric space, were introduced in a previous article by the author. In this article, we present new examples of hyperfinite knots…

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 use Lee's work on the Khovanov homology to define a knot invariant s. We show that s(K) is a concordance invariant and that it provides a lower bound for the slice genus of K. As a corollary, we give a purely combinatorial proof of the…

Let $u(K)$ and $g(K)$ denote the unknotting number and the genus of a knot $K$, respectively. For a 3-braid knot $K$, we show that $u(K)\le g(K)$ holds, and that if $u(K)=g(K)$ then $K$ is either a 2-braid knot, a connected sum of two…

In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong…

We give a sufficient condition under which vanishing property of Cochran-Orr-Teichner knot concordance obstructions splits under connected sum. The condition is described in terms of self-annihilating submodules with respect to higher-order…

In this paper we prove that the Casson-Gordon invariants of the connected sum of two knots split when the Alexander polynomials of the knots are coprime. As one application, for any knot K, all but finitely many algebraically slice twisted…

We give necessary conditions of a surface-knot to be ribbon concordant to another, by introducing a new variant of the cocycle invariant of surface-knots in addition to using the invariant already known. We demonstrate that twist-spins of…

The aim of this survey article is to highlight several notoriously intractable problems about knots and links, as well as to provide a brief discussion of what is known about them.

Piecewise-linear virtual knots are discussed and classified up to edge index six.

Using Gauge theoretical techniques employed by Lisca for 2-bridge knots and by Greene-Jabuka for 3-stranded pretzel knots, we show that no member of the family of Montesinos knots M(0;[m_1+1,n_1+2],[m_2+1,n_2+2],q), with certain…

The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all…

A virtual knot, which is one of generalizations of knots in $\mathbb{R}^{3}$ (or $S^{3}$), is, roughly speaking, an embedded circle in thickened surface $S_{g} \times I$. In this talk we will discuss about knots in 3 dimensional $S_{g}…

Determining when two knots are equivalent (more precisely isotopic) is a fundamental problem in topology. Here we formulate this problem in terms of Predicate Calculus, using the formulation of knots in terms of braids and some basic…

For a knot $K$ in the 3-sphere and a simply connected closed 4-manifold $X$, we define the $X$-double slice genus of $K$, extending the notion from the case when $X$ is the 4-sphere. We show that for each integer $n$, there exists an…

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 paper we show how to realize all knot (and link) types as C^{2} smooth curves of constant curvature. Our proof is constructive: we build the knots with copies of a fixed finite number of "building blocks" that are particular…

We prove a recent conjecture of Manolescu-Willis which states that the $s$-invariant of a knot in $\mathbb{RP}^3$ (as defined by them) gives a lower bound on its null-homologous slice genus in the unit disk bundle of $TS^2$. We also…

We introduce the notion of wen knots, and prove that the set of wen knots is a proper subset of the set of extended welded knots. Furthermore we prove that the complementary subset consists of welded knots up to horizontal mirror…

Given a simply-connected 4-manifold with boundary the 3-sphere, this paper establishes sufficient conditions for a knot in the boundary to be sliced by a locally flat disc in the 4-manifold, whose complement has finite cyclic fundamental…