Related papers: Trefoils and hexafoils in 3-braids
We partially determine grid homology (combinatorial knot Floer homology) of diagonal knots, which are conjectured to be equivalent to positive braid knots, by exploiting nice grid diagrams. Its next-to-top term detects the number of prime…
The goal of this paper is to find a close to isomorphic presentation of 3-manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of…
We show that the triple-crossing number of any knot is greater or equal to twice its (canonical) genus and we show an even stronger bound in the case of links. As an application we show that this bound is strong enough to obtain the…
We study an interacting single-level quantum dot weakly coupled to three electrodes. When two electrodes are biased by voltages with opposite polarities, while keeping the third lead (the stem) grounded, the current through the stem is a…
A torus-covering $T^2$-link of degree $n$ is a surface-link consisting of tori, in the form of an unbranched covering of degree $n$ over the standard torus. We focus on a torus-covering $T^2$-link of degree 3, which is determined by a pair…
The absence of global symmetries in a quantum gravity theory often requires the introduction of (new) symmetry-breaking defects, which appear as singular objects in the low-energy description. This has been formalized in the Cobordism…
A closed geodesic on the modular surface gives rise to a knot on the 3-sphere with a trefoil knot removed, and one can compute the linking number of such a knot with the trefoil knot. We show that, when ordered by their length, the set of…
Motivated by physical and topological applications, we study representations of the group $\mathcal{LB}_3$ of motions of $3$ unlinked oriented circles in $\mathbb{R}^3$. Our point of view is to regard the three strand braid group…
We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe one-handle attachements. As opposed to the usual link calculi of Kirby and others this description uses only…
The notion of a braid is generalized into two and three dimensions. Two-dimensional braids are described by braid monodromies or graphics called charts. In this paper we introduce the notion of curtains, and show that three-dimensional…
Many well studied knots can be realized as positive braid knots where the braid word contains a positive full twist; we say that such knots are twist positive. Some important families of knots are twist positive, including torus knots,…
A conjecture, known as the Shokurov-Koll\'ar connectedness principle, predicts the following. Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-(K_X + B)$ nef over $S$; then, for any point $s \in S$, the…
We compute Cayley graphs and automorphism groups for all finite $n$-quandles of two-bridge and torus knots and links, as well as torus links with an axis.
We consider toroidal compactifications of bosonic string theory with particular regard to the phases (cocycles) necessary for a consistent definition of the vertex operators, the boundary states and the T-duality rules. We use these…
Cobordism of virtual string links on $n$ strands is a combinatorial generalization of link cobordism. There exists a bijection between virtual string links up to cobordisms and elements of the group $\mathbb{Z}^{n(n-1)}$. This paper also…
We show that there are only finitely many homogeneous links whose Conway polynomial has any given degree. Using this we give an example of an inhomogeneous, fibred knot. Secondly, we show how to compute the monodromy of a homogeneous link…
In this paper, we construct invariants of braids, knots and links by studying dynamics of points in $\R^{2}$ and applying the Ptolemy relation $ac+bd=xy$.
A companion paper to "On knot Floer homology in branched double covers" applied to braided branched loci. We reprove the main result of that paper concerning alternating branched loci when projected to an annulus, without using Khovanov…
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…
We prove the conjecture of Przytycki and Sazdanovic that the Khovanov homology of the closure of a 3-stranded braid only contains torsion of order 2. This conjecture has been known for six out of seven classes in the Murasugi-classification…