Computes a specific homology for a type of braid.
problem Calculating a homology for a particular class of braids.
method Uses Khovanov-Rozansky homology for Coxeter braids on 4 strands.
result Computed the triply graded Khovanov-Rozansky homology.
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48 results for “4-strand braids”
New findings on Jones polynomial for 4-strand braids.
problem Whether there are non-trivial knots with trivial Jones polynomial.
method Study of 4-strand braids, exploration of various properties of hypothetical HOMFLY-PT polynomials.
result Existence of a 1-parameter family of 2-variable polynomials that can be HOMFLY-PT polynomials of some knots.
This article is dedicate to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group . Consequently we describe as HNN-extension. As an application to classical braids, we find a new presentation of the Artin pure braid group in terms of the cabled gene…
We show that every trivial 3-strand braid diagram contains a disk, defined as a ribbon ending in opposed crossings. Under a convenient algebraic form, the result extends to every Artin--Tits group of dihedral type, but it fails to extend to braids with 4 strands and more. The proof uses a partition of the Cayley graph …
Proves Burau representation is faithful for 4-strand braids.
problem Proving the Burau representation is faithful for 4-strand braids.
method Used ideas from Moody, Long, Long-Paton, and Bigelow.
result Burau representation is faithful for n = 4.
We give a complete classification of homomorphisms from the braid group on strands to the braid group on strands when is at least 5. We also classify endomorphisms of the braid group on 4 strands, as well as homomorphisms from the commutator subgroup of the braid group on strands to the braid group on …
For at least 7 and equal to 5, we give generating sets of size 2 for the commutator subgroup of the braid group on strands. These generating sets are of the smallest possible cardinality. For equal to 4 or 6, we give generating sets of size three. We also prove that the commutator subgroup of the braid …
We use some Lie group theory and Budney's unitarization of the Lawrence-Krammer representation, to prove that for generic parameters of definite form the image of the representation (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, …
We show that there are contact 3-manifolds of support genus one which admit infinitely many Stein fillings, but do not admit arbitrarily large ones. These Stein fillings arise from genus-1 allowable Lefschetz fibrations with distinct homology groups, all filling a fixed minimal genus open book supporting the boundary c…
New insights into Khovanov homology complexity and topological structure.
problem Complexity of computing Khovanov homology for closed braids.
method Analysis of independence simplicial complexes and polynomial time algorithms.
result Independence simplicial complexes associated to 4-braid diagrams are homotopy equivalent to wedges of spheres.
We derive a factorization of the Alexander polynomial of the 4-strand Turk's head knot using hypergeometric representations.
problem Deriving a factorization of the Alexander polynomial of the 4-strand Turk's head knot
method Using the reduced Burau representation and multivariable resultant elimination over reciprocal constraints
result Deriving a factorization of the Alexander polynomial in terms of Chebyshev polynomials
Injections of Artin groupsmath.GR
We study those Artin groups which, modulo their centers, are finite index subgroups of the mapping class group of a sphere with at least 5 punctures. In particular, we show that any injective homomorphism between these groups is parameterized by a homeomorphism of a punctured sphere together with a map to the integers.…
Algorithm simplifies Khovanov homology computations for 4-strand torus links.
problem Computing Khovanov homology of 4-strand torus links.
method Algorithmic Morse theoretic simplifications postponed until the end.
result Non-trivial Khovanov homology groups in all homological degrees for 4-strand torus links.
Let p and q be distinct integers greater than one. We show that the 2-component pretzel link P(p,q,-p,-q) is not slice, even though it has a ribbon mutant, by using 3-fold branched covers and an obstruction based on Donaldson's diagonalization theorem. As a consequence, we prove the slice-ribbon conjecture for 4-strand…
We prove that the signature bound for the topological 4-genus of 3-strand torus knots is sharp, using McCoy's twisting method. We also show that the bound is off by at most 1 for 4-strand and 6-strand torus knots, and improve the upper bound on the asymptotic ratio between the topological 4-genus and the Seifert genus …
Circle graph complexes reveal link properties via Khovanov homology.
problem Understanding topological properties of links using circle graph complexes.
method Analyzing homotopy types and computing Khovanov homology of specific knots.
result Real extreme Khovanov homology of a 4-strand pretzel knot computed.
The family of negative torus links over a fixed number of strands admits a stable limit in reduced Khovanov homology as grows to infinity. In this paper, we endow this stable space with a bi-graded commutative algebra structure. We describe these algebras explicitly for . As an application, w…
It is known that there are 21 ribbon knots with 10 crossings or fewer. We show that for every ribbon knot, there exists a tangle that satisfies two properties associated with the knot. First, under a specific closure, the closed tangle is equivalent to its corresponding knot. Second, under a different closure, the clos…
Using an obstruction based on Donaldson's theorem on the intersection forms of definite 4-manifolds, we determine which connected sums of lens spaces smoothly embed in S^4. We also find constraints on the Seifert invariants of Seifert 3-manifolds which embed in S^4 when either the base orbifold is non-orientable or the…
The paper computes a knot's Kauffman bracket polynomial using recursive concatenation of a 4-tangle shadow.
problem Computing the Kauffman bracket polynomial for complex knots.
method Recursive concatenation of a 4-tangle shadow, followed by a closure operation and polynomial computation.
result A method to compute the Kauffman bracket polynomial for knots formed from 4-tangle shadows.
Researchers derived Kauffman bracket polynomial for Celtic link shadows using two methods.
problem Calculating the Kauffman bracket polynomial for Celtic link shadows.
method Two complementary approaches: recursive relation and 4-tangle algebra.
result Derived Kauffman bracket polynomial for shadows.
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.
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 -bridge braids.
problem Finding a formula for the braid index of -bridge braids.
method Elementary, effective, self-contained proof.
result Closed form formula for braid index of -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…
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…
Study shows torsion in knot module for specific Montesinos knots.
problem Torsion in Kauffman bracket skein module of Montesinos knots.
method Analyzes Kauffman bracket skein module over for specific knots.
result Provides a negative answer to Problem 1.92 in Kirby's list.
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.
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.
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.
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.
Virtual Braidsmath.GT
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.
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.
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.
problem Understanding the structure of braid monodromy groups of polynomials.
method Analyzing the critical values of polynomials to determine their braid monodromy groups.
result The braid monodromy group of a polynomial equals the braid group if the polynomial has distinct critical values.
Invariants defined for braid systems under Hurwitz equivalence.
problem Invariants for braid systems under Hurwitz equivalence.
method Crossing matrix and polynomial invariant introduced.
result Invariant defined for surface braids and surface links.
Paper defines generalized braids and proves their subgroup status.
problem Understanding the structure of generalized braids and knots.
method Defined generalized braid theories and computed their generating sets.
result Quasitoric normal generalized braids form a subgroup of normal generalized braid group.
Multi-agent Q-learning untangles braids, improving over training.
problem Untangling braids using reinforcement learning.
method Two competing players learn to tangle and untangle braids in the OpenAI Gym environment.
result The system improves at untangling braids with more training, producing good tangled examples.
New family of braided Thompson groups introduced using recursive braids.
problem Dehornoy-Brin braided Thompson group generalization.
method Using recursive braids and strand diagrams to define new groups.
result New groups are finitely generated if is finitely generated.
Paper explores relations between braid groups and their quotients.
problem Understanding relations between braid groups and their quotients.
method Recalling and introducing elements of congruence braid groups, establishing isomorphisms between crystallographic and congruence braid groups.
result Established isomorphisms between crystallographic braid groups and quotients of congruence braid groups.
The Helon model identifies Standard Model quarks and leptons with certain framed braids joined together at both ends by a connecting node (disk). These surfaces with boundary are called braided 3-belts (or simply belts). Twisting and braiding of ribbons composing braided 3-belts are interchangeable, and it was shown in…