Related papers: Tied Links in various topological settings
Given a representation of a link group, we introduce a trilinear form, as a topological invariant. We show that, if the link is either hyperbolic or a knot with malnormality, then the trilinear form equals the pairing of the (twisted)…
Let $\H_g$ be a genus $g$ handlebody and $\MCG_{2n}(\T_g)$ be the $2n$-punctured mapping class group of $\T_g=\partial\H_g$. In this paper we study two particular subgroups of $\MCG_{2n}(\T_g)$ which generalize Hilden groups. As well as…
In a topological description of elementary matter proposed by Bilson-Thompson, the leptons and quarks of a single generation, together with the electroweak gauge bosons, are represented as elements of the framed braid group of three…
The aim of this paper is to realise the techniques of picture-valued invariants and invariants valued in free groups for long knots in the full torus. Such knots and links are of a particular interest because of their relation to Legendrian…
Given a knot or link in the handlebody, $H_g$, of genus $g$ we prove that it can always be represented as the plat closure of a braid in $H_g$. We further establish the Hilden braid group for the handlebody, as a subgroup of the mixed braid…
The main purpose of this article is to classify contact structures on some 3-manifolds, namely lens spaces, most torus bundles over a circle, the solid torus, and the thickened torus T^2 x [0,1]. This classification completes earlier work…
Martin showed that link Floer homology detects braid axes. In this paper we extend this result to give a topological characterisation of links which are almost braided from the point of view of link Floer homology. The result is inspired by…
This is the first paper in a series of three devoted to studying twisted linking forms of knots and three-manifolds. Its function is to provide the algebraic foundations for the next two papers by describing how to define and calculate…
In a recent paper, McMullen showed an inequality between the Thurston norm and the Alexander norm of a 3-manifold. This generalizes the well-known fact that twice the genus of a knot is bounded from below by the degree of the Alexander…
Given a closed, oriented, connected 3-manifold, M, we define higher-order linking forms on the higher-order Alexander modules of M. These higher-order linking forms generalize similar linking forms for knots previously studied by the…
In the paper we give a survey on braid groups and subjects connected with them. We start with the initial definition, then we give several interpretations as well as several presentations of these groups. Burau presentation for the pure…
M. Freedman and V. Krushkal introduced the notion of a "filling" link in a 3-manifold: a link $L$ is filling in $M$ if for any spine $G$ of $M$ disjoint from $L$, $\pi_1(G)$ injects into $\pi_1(M \setminus L )$. Freedman and Krushkal show…
We consider closed acylindrical surfaces in 3-manifolds and in knot and link complements, and show that the genus of these surfaces is bounded linearly by the number of tetrahedra in the triangulation of the manifold and by the number of…
In 2014 Jones showed how to associate links in the $3$-sphere to elements of Thompson's group $F$. We provide an analogue of this program for annular links and Thompson's group $T$. The main result is that any edge-signed graph embedded in…
Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between…
We present new families of examples of non-simple prime Legendrian and transversal knots in tight Lens spaces, which demonstrate that the botany of Legendrians in Lens space is rich. In fact, there are more non-isotopic Legendrians that are…
The concepts of twisted knot theory and singular knot theory inspire the introduction of singular twisted knot theory. This study showcases similar findings for singular twisted links, including the Alexander theorem and the Markov theorem…
In this paper we apply the twisted Alexander polynomial to study the fibering and genus detecting problems for oriented links. In particular we generalize a conjecture of Dunfield, Friedl and Jackson on the torsion polynomial of hyperbolic…
In this paper, we define real link Floer homology for strongly invertible and doubly periodic links in closed real $3$-manifolds with connected fixed sets, which generalizes real Heegaard Floer homology and real sutured Heegaard Floer…
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…