arXiv research

Multi-agent Q-learning untangles braids, improving over training.

Neural nets solve braid untangling up to length 20.

Motivated by the work in [15], this paper deals with the theory of the braids from chromatic configuration spaces. This kind of braids possess the property that some strings of each braid may intersect together and can also be untangled, so they are quite different from the ordinary braids in the sense of Artin. This e…

Positive braids minimize knot untangling steps.

Study on bounds of knot untangling for specific types of knots.

Study shows fibred knots can't be untied with specific moves.

It has long been known to mathematicians and physicists that while a full rotation in three-dimensional Euclidean space causes tangling, two rotations can be untangled. Formally, an untangling is a based nullhomotopy of the double-twist loop in the special orthogonal group of rotations. We study a particularly simple, …

The paper analyzes the complexity of untangling knots with a given number of moves.

In mathematics, a knot is a single strand of string crossed over itself any number of times, and connected at the ends. The Reidemeister Moves have been proven to be the three core moves necessary to fully untangle a knot. Some knots can be untangled to a loop (the unknot), while others are fundamentally knotted. We de…

Paper introduces untangling number to quantify 3-periodic tangle complexity.

We introduce and illustrate a new approach to the unknotting problem via the dynamics of vortex strings in a nonlinear partial differential equation of reaction-diffusion type. To untangle a given knot, a Biot-Savart construction is used to initialize the knot as a vortex string in the FitzHugh-Nagumo equation. Remarka…

Paper defines untangling number to measure entanglement complexity in 3-periodic networks.

We introduce HUBERT which combines the structured-representational power of Tensor-Product Representations (TPRs) and BERT, a pre-trained bidirectional Transformer language model. We show that there is shared structure between different NLP datasets that HUBERT, but not BERT, is able to learn and leverage. We validate …

Motivated by the results of Scott and Patel about "untangling" closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back …

It is a major unsolved problem as to whether unknot recognition - that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle - has a polynomial time algorithm. In practice, trivial knots (which can be untangled) are typically easy to identify using fast simplification techniques, wherea…

Knot 11n102 requires 2 changes to untangle.

New moves help untangle complex knots.

Robot untangles knots by walking and switching crossings.

We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in $S^2$ and in $\R^2$.

Let $K$ be a link of Conway's normal form $C(m)$, $m \geq 0$, or $C(m,n)$ with $mn\textgreater{}0$, and let $D$ be a trigonal diagram of $K.$ We show that it is possible to transform $D$ into an alternating trigonal diagram, so that all intermediate diagrams remain trigonal, and the number of crossings never increases.

Study contact instantons and Legendrian links, proving energy inequalities.

Improved tracking of tangled point sources using Riemannian metrics.

We consider two systems of curves $(α_1,...,α_m)$ and $(β_1,...,β_n)$ drawn on a compact two-dimensional surface $M$ with boundary. Each $α_i$ and each $β_j$ is either an arc meeting the boundary of $M$ at its two endpoints, or a closed curve. The $α_i$ are pairwise disjoint except for possibly sharing endpoints, and s…

We show that the driving force behind the regularizing effect of Laplacian smoothing on surface elements is the popular mean ratio quality measure. We use these insights to provide natural generalizations to polygons and polyhedra. The corresponding functions measuring the quality of meshes are easily seen to be convex…

We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard, including detecting whether a link contains a trivial sublink with $n$ componen…

Classic braids embed in virtual braids.

The paper finds minimal generating sets and abelianizes the quasitoric braid group.

We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words…

Satellite links of fully positive braids are characterized.

Formula found for braid index of $n$-bridge braids.

Most simple braids have positive topological entropy.

This paper is concerned with detecting when a closed braid and its axis are 'mutually braided' in the sense of Rudolph. It deals with closed braids which are fibred links, the simplest case being closed braids which present the unknot. The geometric condition for mutual braiding refers to the existence of a close contr…

Suppose $K$ is an unknot lying in the 1-skeleton of a triangulated 3-manifold with $t$ tetrahedra. Hass and Lagarias showed there is an upper bound, depending only on $t$, for the minimal number of elementary moves to untangle $K$. We give a simpler proof, utilizing a normal form for surfaces whose boundary is containe…

We show that 3-braid links with given (non-zero) Alexander or Jones polynomial are finitely many, and can be effectively determined. We classify among closed 3-braids strongly quasipositive and fibered ones, and show that 3-braid links have a unique incompressible Seifert surface. We also classify the positive braid wo…

The paper examines subgroup separability for surface and virtual braid groups.

Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility, stability and Reidemeister moves. We show that virtual braids are in a bijective…

Study on deformation cohomology for braided commutative structures.

Magic braids can be made with leatherworking technique.

Study virtual braid groups, proving a key subgroup result.

Machine learning classifies braids and discovers new invariants.

In the present paper we give a new method for converting virtual knots and links to virtual braids. Indeed the braiding method given in this paper is quite general, and applies to all the categories in which braiding can be accomplished. We give a unifying topological interpretation of virtuals and flats (virtual strin…

This paper extends braid lifting to coloured braid groupoids for all simple disc covers.

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 braids are described by braid monodromies or curtains.

New braid representations using virtual knot theory.

Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer g…

Polynomials with distinct critical values have braid monodromy groups equal to braid groups.

Invariants defined for braid systems under Hurwitz equivalence.

Paper defines generalized braids and proves their subgroup status.