Study on groups of unrestricted virtual braids and their properties.
problem Understanding the structure and properties of virtual braids and links.
method Exploration of unrestricted virtual braids, fused links, flat virtual braids, and virtual Gauss braids.
result Definition and study of groups of flat virtual braids and virtual Gauss braids, including their linearity.
Study shows knots from certain braids are algebraically slice under specific conditions.
problem Characterizing knots formed by specific braids.
method Gauss diagram formulas to analyze algebraic sliceness.
result Knots from βn are algebraically slice if and only if n is odd. Paper explores algebraic, topological, and combinatorial properties of singular virtual braids.
problem Understanding singular virtual braids and their properties.
method Algebraic relations, topological and combinatorial bijections, presentations.
result A bijection between singular abstract braids and singular virtual braids, leading to a presentation of the singular pure virtual braid monoid.
A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we present simple formulas for an infinite family of invariants in terms of counting su…
New polynomial invariants detect non-invertibility of closed braids.
problem Detecting non-invertibility of closed braids.
method Polynomial 1-cocycles evaluated on closed braids' full rotation.
result Polynomial invariants can detect non-invertibility of 2-component links.
The paper defines new representations and groups related to virtual links.
problem Defining and studying new representations of virtual braid groups and link groups.
method Introducing virtually symmetric representations, virtual link groups, and marked Gauss diagrams.
result Established equivalence of many known representations to virtually symmetric representations.
We define a new kind of Gauss diagrams to describe knots in the solid torus with projections in the annulus. We see that it provides an efficient tool for showing that a knot diagram can be fully recovered from its decorated Gauss diagram, and we use it to establish a characterization of the decorated Gauss diagrams of…
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…
We found a way to code meanders and show they are idempotent.
problem Understanding and coding meandric permutations.
method We established a bijection between meanders and Gauss diagrams, and used this to construct matrices that are idempotent.
result Meandric permutations are idempotent over the field GF(2).
Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. Virtual braids correspond naturally to virtual knots. We consider the group of virtual braids on n strings VB_n and its Burau representation, in particular we study their homological properties. We prove that the plus-constructi…
The paper characterizes crystallographic groups derived from virtual braid and twin groups.
problem Characterizing crystallographic groups from virtual braid and twin groups.
method Analyzing quotients of virtual braid and twin groups by their commutator subgroups.
result The quotients of virtual braid and twin groups by their commutator subgroups are crystallographic groups.
Goussarov, Polyak, and Viro proved that finite type invariants of knots are ``finitely multi-local'', meaning that on a knot diagram, sums of quantities, defined by local information, determine the value of the knot invariant. The result implies the existence of Gauss diagram combinatorial formulas for finite type inva…
We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or self-virtualizations. We provide a number of results which point out the differences between these various notions. Th…
We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allo…
A systematic description of the Wess-Zumino-Witten model is presented. The symplectic method plays the major role in this paper and also gives the relationship between the WZW model and the Chern-Simons model. The quantum theory is obtained to give the projective representation of the Loop group. The Gauss constraints …
New formulas classify higher-dimensional knots and links.
problem Classifying smooth embeddings of (2ℓ−1)-spheres into R3ℓ. method Similar to Goussarov-Polyak-Viro, project higher-dimensional knots onto a hyperplane and study double and singular points.
result Obtained combinatorial formulas for invariants of smooth embeddings of (4k−1)-dimensional knots and links in R6k. Using the recent Gauss diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of …
Classic braids embed in virtual braids.
problem Embedding classic braids in virtual braids.
method Elementary proof using generalizations of braid groups.
result Classical braid group injects into virtual braid group.
Gauss codes help uniquely identify virtual doodles.
problem Identifying virtual doodles uniquely.
method Introduced left canonical Gauss codes.
result Oriented virtual doodles uniquely presented by left canonical Gauss codes.
Study minimal surfaces in 4D, find specific tori with total curvature -8π.
problem Find complete proper non-holomorphic minimal tori in R^4 with total curvature -8π.
method Use tools like Gauss maps and link/braid/writhe at infinity. Translate problem into a system of equations involving Weierstrass function. Explicitly solve for rectangular and square tori.
result Explicit solutions for minimal tori in R^4, including generalization of Chen-Gackstetter torus in R^3.
The paper finds minimal generating sets and abelianizes the quasitoric braid group.
problem Understanding the structure of quasitoric braids and their subgroup properties.
method Provided two minimal generating sets and determined the abelianization.
result Minimal generating sets and abelianization of the quasitoric braid group were determined.
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.
problem Characterizing satellite links of fully positive braids.
method Analyzing fully positive braids and their satellites.
result Satellite links of fully positive braids are characterized by specific conditions.
Formula found for braid index of n-bridge braids.
problem Finding a formula for the braid index of n-bridge braids. method Elementary, effective, self-contained proof.
result Closed form formula for braid index of n-bridge braids. Most simple braids have positive topological entropy.
problem Understanding the topological entropy of simple braids.
method Reduction from simple braids to non-simple 3-strand braids.
result The proportion of simple braids with positive entropy approaches 100% as the number of strands increases.
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…
Gauss diagrams' properties can change with Hamiltonian cycle choice.
problem The impact of Hamiltonian cycle choice on Gauss diagrams.
method Examined realizable and unrealizable Gauss diagrams, and proved preservation of realizability under certain Hamiltonian cycle changes.
result Properties of Gauss diagrams can vary with Hamiltonian cycle choice.
Paper proves Markov's theorem for extended welded braids and links.
problem Generalizing welded links to extended welded links and braids.
method Following Kamada's approach, proving Alexander and Markov's theorems for extended welded braids and links.
result Proves versions of Alexander and Markov's theorems for extended welded braids and links.
By defining combinatorial moves, we can define an equivalence relation on Gauss words called homotopy. In this paper we define a homotopy invariant of Gauss words. We use this to show that there exist Gauss words that are not homotopically equivalent to the empty Gauss word, disproving a conjecture by Turaev. In fact, …
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.
problem Subgroup separability of surface and virtual braid groups.
method Study of subgroup separability (LERF) properties.
result Properties of subgroup separability for surface and virtual braid groups are explored.
Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.
Algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
problem Recognizing and performing Reidemeister moves in Gauss diagrams.
method Simple algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
result Simple algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
Polynomial-time algorithm for virtual braid triviality.
problem Determining if virtual braid words represent the trivial braid.
method An O(l3n)-time algorithm for VBn. result Determines triviality of virtual braid words efficiently.
This paper identifies braided 3-belts that can be written in a braid-only form.
problem Identifying braided 3-belts that can be written in a braid-only form.
method Developed an algorithm to calculate the braid word for braided 3-belts and determined the conditions for knotted boundaries.
result Identified the set of braided 3-belts that can be written in a braid-only form and derived a formula for the Jones polynomial for knotted boundaries.
Magic braids can be made with leatherworking technique.
problem Determine which braids can be made with leatherworking technique.
method Explore their relation to several braid groups.
result Magic braids can be made with leatherworking technique.
Study virtual braid groups, proving a key subgroup result.
problem Understanding congruence subgroups in virtual braid groups.
method Using an extension of the integral Burau representation.
result Proved level 2 congruence subgroup of virtual braid group is pure virtual braid group.
Machine learning classifies braids and discovers new invariants.
problem Classifying and discovering invariants of braids and flat braids.
method Supervised learning with neural networks to classify braids as trivial or non-trivial.
result Found new convenient invariants of braids, including a complete invariant of flat braids.
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.
problem Lifting braids to homeomorphisms on branched covers of the disc.
method Defines a map from a coloured braid groupoid to a mapping class groupoid for all simple covers of the disc.
result Characterizes the lift of every coloured braid, recovering classical lifting on liftable braids.
Paper studies special braids from chromatic configuration spaces.
problem Understanding braids with crossing-changeable properties.
method Theory of chromatic configuration spaces and braids.
result Introduced braids with the ability to intersect and untangle.
The paper explores various braid-like groups and their properties.
problem Investigating the pure braid groups and their relatives.
method Examining resonance varieties, lower central series ranks, Chen ranks, residual and formality properties.
result Discussed natural homomorphisms and methods to distinguish braid-like groups.
New virtual braid group structure and presentation found.
problem Understanding the structure of virtual pure braid groups.
method Cabling construction and HNN-extension approach.
result New presentation of VP4 and P4 groups. New concept of boundary braids defined for disk configurations.
problem Defining and studying braids with points fixed on the boundary.
method Using configuration spaces and fundamental groups, defining boundary braids and analyzing their geometric properties.
result Boundary braids form a subcomplex that metrically splits into a Euclidean polyhedron and a smaller rank dual braid complex.
Paper discusses isometric immersion of surfaces with slowly decaying negative Gauss curvature.
problem Isometric immersion of complete surfaces with slowly decaying negative Gauss curvature.
method Gauss-Codazzi system, novel observations on decay properties of Riemann invariants, weighted Riemann invariants, comparison principle.
result Surface has a global smooth isometric immersion in three-dimensional Euclidean space if Gauss curvature decays slowly at infinity.
Incorrect parity-based descriptions of realizable Gauss diagrams found, but bipartite graphs provide a valid approach.
problem Incorrect descriptions of realizable Gauss diagrams using parity conditions.
method Used bipartite graphs to describe realizable Gauss diagrams.
result Realizable Gauss diagrams can be accurately described using bipartite graphs.
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.
problem No classical features in virtual knot theory.
method Construct new braid representations using virtual knot theory.
result New representations of classical braids.