arXiv research

A new method for describing surface-links in 4-space.

Paper computes knot symmetric quandle for surface-links and finds infinitely many distinct surface-knots.

New methods convert complex link presentations to simpler, recognizable forms.

Paper explores link and plat presentations, showing equivalence under bridge isotopy.

Paper proves unique canonical form for certain highly twisted knots and links.

In this paper, a method is given to calculate the Jones polynomial of the 6-plat presentations of knots by using a representation of the braid group $\mathbb{B}_6$ into a group of $5\times 5$ matrices. We also can calculate the Jones polynomial of the $2n$-plat presentations of knots by generalizing the method for the …

Simplified plat diagrams for unlink without stabilization.

Proves knots in handlebodies can be represented as plats of braids.

Algorithm converts plat to standard closure of braids in 3D and related spaces.

The paper defines plat closures for spherical braids and shows links in $\mathbb{R}P^3$ can be realized this way.

Defines a strict order on plat presentation classes for links.

This paper classifies 2-plat 2-knots using a new invariant.

The present work is devoted to compact completely solvable solvmanifolds which admit Kahlerian metrics whose Kahler forms are homogeneous. In particular, we show that such manifolds are diffeomorphic to flat tori. Our proof is based on Dynkin diagrams associated to left invariant closed 2-forms in completely solvable L…

We characterize which Legendrian $4$-plat knots in the standard contact $3$-space have exact orientable Lagrangian fillings. As a corollary, we show that the underlying smooth knot types of fillable Legendrian $4$-plats are positive.

The article finds equivalence moves for links in specific manifolds using plat closure of braids.

Every knot has a plat projection, obtained by closing up a braid with bridges. The plat projection is determined by the number of strands and the number of rows of twist regions in the braid, and an integer number of crossings in each twist region. In recent work, we showed that under certain restrictions, including th…

The paper constructs infinitely many prime hyperbolic knots.

New proof shows all knots in certain plat diagrams are hyperbolic.

New 'book links' generalize braids and plats, proving Markov's theorem.

We calculate the bridge distance for $m$-bridge knots/links in the $3$-sphere with sufficiently complicated $2m$-plat projections. In particular we show that if the underlying braid of the plat has $n - 1$ rows of twists and all its exponents have absolute value greater than or equal to three then the distance of the b…

Un sous-système de dimension différentielle au plus 2 d'une extension plate est plate. Si un tel système plat est stationnaire, il admet des sorties plates indépendantes du temps. A subsystem of a flat system of differential dimension at most 2 is flat. Furthermore, if such a flat system is stationary, we show that the…

Algorithm finds plat-equivalence words for genus 2 3-manifolds.

New method uses non-orientable surfaces to describe knots in 3-manifolds.

Characterizes knot groups and symmetric quandles of surface-links.

The paper studies knitted surfaces and surface-links, showing their isotopy and closure properties.

We prove a Markov theorem for tame links in a connected closed orientable 3-manifold $M$ with respect to a plat-like representation. More precisely, given a genus $g$ Heegaard surface $Σ_g$ for $M$ we represent each link in $M$ as the plat closure of a braid in the surface braid group $B_{g,2n}=π_1(C_{2n}(Σ_g))$ and an…

In this paper, I give a method to calculate the HOMFLY polynomials of knots by using a representation of the braid group B4 into a group of 3 ? 3 matrices. Also, I will give examples of a 2-bridge knot and a 3-bridge knot that have the same Jone polynomial, but different HOMFLY polynomials.

We give a new definition of the knot invariant associated to the Lie algebra su_{N+1}. The knot or link must be presented as the plat closure of a braid. The invariant is then a homological intersection pairing between two submanifolds of a configuration space of points in a disk. This generalizes previous work on the …

New framed moves extend classical knot theory results.

We extend the tangle model, originally developed by Ernst and Sumners, to include composite knots. We show that, for any prime tangle, there are no rational tangle attachments of distance greater than one that first yield a 4-plat and then a connected sum of 4-plats. This is done by building on results on exceptional D…

We show, by an elementary and explicit construction, that the group of Hamiltonian diffeomorphisms of certain symplectic manifolds, endowed with Hofer's metric, contains subgroups quasi-isometric to Euclidean spaces of arbitrary dimension.

We generalize a theorem of Finkelstein and Moriah and show that if a link $L$ has a $2n$-plat projection satisfying certain conditions, then its complement contains some closed essential surfaces. In most cases these surfaces remain essential after any totally nontrivial surgery on $L$.

We study a certain type of braid closure which resembles the plat closure but has certain advantages; for example, it maps pure braids to knots. The main results of this note are a Markov-type theorem and a description of how Vassiliev invariants behave under this braid closure.

Ozsvath and Szabo recently constructed an algebraically defined invariant of tangles which takes the form of a DA bimodule. This invariant is expected to compute knot Floer homology. The authors have a similar construction for open braids and their plat closures which can be viewed as a filtered DA bimodule over the sa…

We show that for an $n$-component, $n$-bridge link and a positive integer $m$, the following is true: If the longitudes of $L$ lie in the $(m+2)$-th term of the lower central series of the link group then all the finite type invariants of orders $\leq m$ for $L$ are the same as these of the $n$-component unlink.

Proof of Knot Entropy Conjecture for tube lattice polygons.

A surprising image of the stock market arises if the price time series of all Dow Jones Industrial Average stock components are represented in one chart at once. The chart evolves into a braid representation of the stock market by taking into account only the crossing of stocks and fixing a convention defining overcros…

Let $M$ be a maximal globally hyperbolic Cauchy compact flat spacetime of dimension 2+1, admitting a Cauchy hypersurface diffeomorphic to a compact hyperbolic manifold. We study the asymptotic behaviour of level sets of quasi-concave time functions on $M$. We give a positive answer to a conjecture of Benedetti and Guad…

We give a new definition of the Jones polynomial. Let L be an oriented knot or link obtained as the plat closure of a braid beta in B_{2n}. We define a covering space tilde{C} of the space of unordered n-tuples of distinct points in the 2n-punctured disk. We then describe two n-manifolds tilde{S} and tilde{T} in tilde{…

Solves double coset problem for braid group H_n.

A tangle is an oriented 1-submanifold of the cylinder whose endpoints lie on the two disks in the boundary of the cylinder. Using an algebraic tool developed by Lescop, we extend the Burau representation of braids to a functor from the category of oriented tangles to the category of Z[t,t^{-1}]-modules. For (1,1)-tangl…

We show that the volumes of certain hyperbolic A-adequate links can be bounded (above and) below in terms of two diagrammatic quantities: the twist number and the number of certain alternating tangles in an A-adequate diagram. We then restrict our attention to plat closures of certain braids, a rich family of links who…

New knot invariant from 3-braids and 6-valent graphs.

The aim of this work is to study the foliations on the complex projective plane with flat \textsc{Legendre} transform (dual web). We establish some effective criteria for the flatness of the dual $d$-web of a homogeneous foliation of degree $d$ and we describe some explicit examples. These results allow us to show that…

[Original abstract (1992):] The modulus of quasipositivity q(K) of a knot K was introduced as a tool in the knot theory of complex plane curves, and can be applied to Legendrian knot theory in symplectic topology. It has also, however, a straightforward characterization in ordinary knot theory: q(K) is the supremum of …

In this paper we introduce a chain complex $C_{1 \pm 1}(D)$ where D is a plat braid diagram for a knot K. This complex is inspired by knot Floer homology, but it the construction is purely algebraic. It is constructed as an oriented cube of resolutions with differential d=d_0+d_1. We show that the E_2 page of the assoc…

New covering moves for 3-manifolds up to degree 4.

It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maxim…