arXiv research

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

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

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

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

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

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

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

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

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.

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…

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

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.

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 …

Solves double coset problem for braid group H_n.

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

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…

Simplified plat diagrams for unlink without stabilization.

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

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{…

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

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…

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 …

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.

We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for appro…

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

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…

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…

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

Defines a strict order on plat presentation classes for links.

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

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 $L$.

Characterizes knot groups and symmetric quandles of surface-links.

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.

Study on knot classification using 3-braid closures and ribbon surfaces.

New proof classifies orbit closures in Hodge bundle.

Proof of Knot Entropy Conjecture for tube lattice polygons.

We study a regular closure operator in the category of quandles. We show that the regular closure operator and the pullback closure operator corresponding to the reflector from the category of quandles to its full subcategory of trivial quandles coincide, we give a simple description of this closure operator, and analy…

New framed moves extend classical knot theory results.

The study connects lamination and orbit closures in hyperbolic manifolds.

Study orbits in right triangles, deducing periodic billiard paths and classifying orbit closures.