Related papers: Link homotopic but not isotopic
Based on some analogies with the Hodge theory of isolated hypersurface singularities, we define Hodge-type numerical invariants (called H-numbers) of any, not necessarily algebraic, link in $S^3$. They contain the same information as the…
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…
To each link $L$ in $S^3$ we associate a collection of certain labelled directed trees, called width trees. We interpret some classical and new topological link invariants in terms of these width trees and show how the geometric structure…
We show that for every m in N, there exists an n in N such that every embedding of the complete graph K_n in R^3 contains a link of two components whose linking number is at least m. Furthermore, there exists an r in N such that every…
We show that if a link L with non-zero Alexander polynomial admits a locally flat cobordism to a `weakly m-split link', then the cobordism must have genus at least (m-1)/2. This generalises a recent result of J. Pardon.
A new classification theorem for links by the authors and Roger Fenn leads to computable link invariants. As an illustration we distinguish the left and right trefoils and recover the result of Carter et al that the 2-twist-spun trefoil is…
Let $L\subset S^3$ be a link. We study the Heegaard Floer homology of the branched double-cover $\Sigma(L)$ of $S^3$, branched along $L$. When $L$ is an alternating link, $\HFa$ of its branched double-cover has a particularly simple form,…
We reconfigure the Milnor invariant of links in terms of central group extensions and unipotent Magnus embeddings. We also develop a diagrammatic computation of the invariant and compute the first non-vanishing invariants of the Milnor link…
In every Engel manifold we construct an infinite family of pairwise non-isotopic transverse tori that are all smoothly isotopic. To distinguish the transverse tori in the family we introduce a homological invariant of transverse tori that…
In this paper we define the 1,2-coloured HOMFLY-PT link homology and prove that it is a link invariant. We conjecture that this homology categorifies the coloured HOMFLY-PT polynomial for links whose components are labelled 1 or 2.
The linking number $lk$ is defined if link components are zero homologous. Our affine linking invariant $alk$ generalizes $lk$ to the case of linked submanifolds with arbitrary homology classes. We apply $alk$ to the study of causality in…
Given positive linear functional l on a vector lattice L of real functions, and a vector subspace M of L, we construct a vector subspace P(M) of M in such a way that 1) l is nullcontinuous on P(M), and 2) if l is nullcontinuous on M then…
A class of one-relator groups such that every group in the class is determined by a triple of integers and is an HNN-extension of some Baumslag -- Solitar group is considered. A criterion for two groups in this class to be isomorphic and…
We introduce a new very large family of transformations of rectangular diagrams of links that preserve the isotopy class of the link. We provide an example when two diagrams of the same complexity are related by such a transformation and…
Given a link $L$, a representation $\pi_1(S^{3}-L)\to {\rm SL}(2,\mathbb{C})$ is {\it tracefree} if the image of each meridian has trace zero. We determine the conjugacy classes of tracefree representations when $L$ is a Montesinos link.
We prove that a graph is intrinsically linked in an arbitrary 3-manifold M if and only if it is intrinsically linked in S^3. Also, assuming the Poincare Conjecture, we prove that a graph is intrinsically knotted in M if and only if it is…
We prove some necessary conditions for a link to be either concordant to a quasi-positive link, quasi-positive, positive, or the closure of a positive braid. The main applications of our results are a characterisation of positive links with…
A thick link is a link in $\mathbb{R}^3$ such that each component of the link lies at distance at least $1$ from every other component. Strengthening the notion of thickness, we define a thickly embedded link to be a thick link whose open…
We give infinitely many $2$-component links with unknotted components which are topologically concordant to the Hopf link, but not smoothly concordant to any $2$-component link with trivial Alexander polynomial. Our examples are pairwise…
We study intrinsically linked graphs where we require that every embedding of the graph contains not just a non-split link, but a link that satisfies some additional property. Examples of properties we address in this paper are: a two…