Motivated by the works of Krasner [arXiv:0801.4018] and Lobb [arXiv:1103.1412], we simplify the Khovanov-Rozansky chain complexes of open 2-braids. As an application, we show that, for a knot containing a "long" 2-braid, the sl(N) Rasmussen invariant of this knot depends linearly on the length of this 2-braid. We refin…
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48 results for “open 2-braids”
New stable homotopy types for knots and diagrams, aiding in their computation.
problem Computing Khovanov-Rozansky cohomology for knots and diagrams.
method Lifted simplified Khovanov-Rozansky complexes to stable homotopy types for open 2-braids.
result New stable homotopy types for knots and diagrams exhibit interesting behavior.
Classifies certain graph 2-braid groups up to quasi-isometry.
problem Classifying 2-braid groups over graphs up to quasi-isometry.
method Using intersection complexes and right-angled Artin groups.
result Classifies 2-braid groups over graphs with circumference ≤ 1 up to quasi-isometry.
We give formulae for the first homology of the -braid group and the pure 2-braid group over a finite graph in terms of graph theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the -braid group over the graph is torsion-free and the conjectures about the first h…
We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relation, leading to six categorified relations. …
Let and denote the unknotting number and the genus of a knot , respectively. For a 3-braid knot , we show that holds, and that if then is either a 2-braid knot, a connected sum of two 2-braid knots, the figure-eight knot, a strongly quasipositive knot or its mirror ima…
We define Dynnikov coordinates on virtual braid groups. We prove that they are faithful invariants of virtual 2-braids, and present evidence that they are also very powerful invariants for general virtual braids.
Study quasi-isometry invariants of square complexes and their applications.
problem Classifying quasi-isometry types of 2D right-angled Artin groups and graph 2-braid groups.
method Define and analyze intersection complexes for universal covers of weakly special square complexes.
result Discover new quasi-isometric relationships between graph 2-braid groups and right-angled Artin groups.
New algebra connects braid group actions to disc curves.
problem Understanding braid group actions on disc curves.
method Constructed a type B zigzag algebra and showed its categorical action on projective modules.
result Type B braid group action on homotopy category of projective modules.
We prove hyperbolicity for links in thickened surfaces.
problem Proving hyperbolicity for links in thickened surfaces.
method Defining fully alternating links and proving their hyperbolicity; extending results to essential surfaces.
result Prime, fully alternating links in are hyperbolic.
Study large-scale geometry of graph braid groups via cubical structures.
problem Classify and understand the quasi-isometry of graph braid groups.
method Exploit cubical structures to relate hyperbolicity, undistorted subgroups, and group decompositions.
result Complete classification of graph braid groups quasi-isometric to free groups.
New links are hyperbolic, with volume computations.
problem Understanding the hyperbolic nature of augmented links.
method Extending the definition of augmented links to include more trivial components and proving hyperbolicity.
result Generalized augmented links are hyperbolic.
We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index . In order to have the necessity part, graphs are organized into small classes so that one of homological or cohomological characteristics of right-angled Artin groups can be applied. Fi…
Algorithm finds plat-equivalence words for genus 2 3-manifolds.
problem Equivalence of links in genus 2 3-manifolds.
method Algorithm implemented in c++ to find plat-equivalence words in braid groups.
result Extends genus 1 results to genus 2, describing words for notable manifolds.
On Marked Braid Groupsmath.GT
Introduces decorated braids for knot and braid theory.
problem Enhance braid theory with additional information at crossings.
method Introduces -braids and -braids, generalizing parity.
result Dotted braid groups are a natural enhancement of classical braid groups.
Non-Reidemeister Knot Theory and Its Applications in Dynamical Systems, Geometry, and Topologymath.GT
Classical knot theory deals with {\em diagrams} and {\em invariants}. By means of horizontal {\em trisecants}, we construct a new theory of classical braids with invariants valued in {\em pictures}. These pictures are closely related to diagrams of the initial object. The main tool is the notion of {\em free -braid …
New findings on hyperbolicity of augmented links in thickened surfaces.
problem Proving hyperbolicity of links in thickened surfaces.
method Extending hyperbolicity results to generalized augmented cellular alternating links.
result Generalized augmented cellular alternating links in thickened surfaces are hyperbolic.
New categorical actions link topological and algebraic structures.
problem Understanding relationships between topological and algebraic structures.
method Categorical actions of type B braid group on homotopy categories.
result Proves Rouquier's conjecture on faithfulness of Type B 2-braid group.
Introduces nested open books for manifold decomposition.
problem Decomposing manifolds using open book structures.
method Defines nested open books and their push-offs, applies to fibre connected sums.
result Explicitly describes open book decompositions of fibre connected sums.
Survey on open manifolds with nonnegative Ricci curvature and open questions.
problem Understanding fundamental groups of open manifolds with nonnegative Ricci curvature.
method Survey and open questions.
result Discussion of open questions in the field.
Coverings of open booksmath.GT
Study open book coverings and their applications in contact geometry.
problem Understanding the structure of open book foliations and their impact on contact geometry.
method Using open book foliations to explore coverings and virtually overtwisted contact manifolds.
result Open book coverings produce transverse knots with depth greater than 1.
Handles decompositions reveal new open book structures.
problem Understanding and manipulating open book structures on manifolds.
method Build handle decompositions and describe handle slides.
result Stabilization of open books to trivial monodromy.
Paper proves non-prime 3-manifolds with specific open book genus have particular prime components.
problem Proving non-prime 3-manifolds with open book genus 2 have non-trivial prime pieces.
method Analyzing open book decompositions and connected sums of 3-manifolds.
result Proves non-prime 3-manifolds with open book genus 2 are homeomorphic to for some integers .
Foliated Open Booksmath.GT
The paper introduces foliated open books for contact 3-manifolds with boundary foliations.
problem Studying contact manifolds with convex boundary using finer tools.
method Developed a new type of open book decomposition with a specified characteristic foliation on the boundary.
result Established the uniqueness and existence of foliated open books and their equivalence to other models.
Embeds 3-manifolds into higher dimensions using open books.
problem Embedding 3-manifolds into higher-dimensional spaces.
method Using open book decompositions of and .
result Every closed orientable 3-manifold embeds in .
The abstract discusses embedding manifolds in open books and contact structures.
problem Embedding manifolds in open books and contact structures.
method Using open book embeddings and contact structures, the abstract proves embedding theorems for various manifolds.
result Closed manifolds can be embedded in open books and contact structures.
Extended Alexander groups are used to define an invariant for open virtual strings. Examples of non-commuting open strings and a ribbon-concordance obstruction are given. An example is given of a slice virtual open string that is not ribbon. Definitions are extended to open n-strings.
In the present paper we describe compatible open books for the fibre connected sum along binding components of open books, as well as for the fibre connected sum along multi-sections of open books. As an application the first description provides simple ways of constructing open books supporting all tight contact struc…
New method shows links can be braided open book bindings.
problem Understanding fibered links and their bindings.
method Mutual arc presentations and braided open books.
result Every fibered link is the binding of a braided open book.
New examples show high twisting doesn't guarantee open book maximality.
problem Maximality of open book pages under high twisting.
method Double branched covers of closed braids.
result Arbitrarily high twisting does not guarantee maximal Euler characteristic.
We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, we describe horizo…
Algorithm constructs Kirby diagrams for 4D open books.
problem Constructing Kirby diagrams for 4D open books.
method Algorithm using Heegaard diagrams of pages.
result Diffeomorphic open books constructed with different pages and monodromies.
First example of open manifold with positive Ricci curvature and non-proper Busemann function.
problem Counterexample to Busemann function properness in open manifolds with nonnegative Ricci curvature.
method Provided an open manifold with positive Ricci curvature and non-proper Busemann function.
result First example of open manifold with positive Ricci curvature and non-proper Busemann function.
Study shows range of simplicial volumes for open manifolds.
problem Understanding simplicial volumes of open manifolds.
method Analyzes locally finite simplicial volumes in dimensions at least 4.
result Set of simplicial volumes is [0, ∞] for open manifolds.
Research improves open-set learning by leveraging unlabelled data.
problem Learning between observed and unobserved novel categories.
method Unified policy of positive and unlabelled learning, semi-supervised learning, and open-set recognition.
result Achieves state-of-the-art results in open-set learning.
Study homotopy groups of open books and their pages, pages, and bindings.
problem Homotopy groups of open books and their components.
method Homotopy theoretic conditions on monodromy, integral and rational loop space decompositions.
result Integral and rational loop space decompositions for open books under specific conditions.
Simplicial volume vanishes for 4-manifolds with open book decompositions.
problem Characterizing open book decompositions in dimension 4.
method Adapted proof by Bucher and Neofytidis.
result Simplicial volume vanishes for 4-manifolds with open book decompositions.
Open problems in billiards and optics from a workshop.
problem Open problems in billiards and geometric optics.
method Collection of open problems from a workshop.
result No specific key result mentioned; collection of problems.
Common positive stabilisation found for isotopic contact structures.
problem Isotopic contact structures and open book decompositions.
method Sequence of positive stabilisations and destabilisations.
result Common positive stabilisation exists for isotopic contact structures.
New proof of Giroux Correspondence for tight contact 3-manifolds.
problem Proving the Giroux Correspondence for tight contact 3-manifolds.
method Introducing tight Heegaard splittings, using refinement process, and translating moves between splittings to moves between open books.
result Proves the tight Giroux Correspondence for contact 3-manifolds.
The authors introduce Morse foliated open books for studying contact manifolds.
problem Studying contact manifolds with boundary.
method Introducing Morse foliated open books and extending the right-veering concept.
result Right-veering plays a similar role in detecting overtwistedness in foliated open books.
Survey of open problems in finite-dimensional integrable systems.
problem Open problems in finite-dimensional integrable systems.
method None specified; survey of existing open problems.
result Many open problems were identified from a conference.
As an application of the construction of open books on plumbed 3-manifolds, we construct elliptic open books on torus bundles over the circle. In certain cases these open books are compatible with Stein fillable contact structures and have minimal genus.
Proves codimension 2 spun embedding for specific manifolds.
problem Codimension 2 spun embedding for manifolds.
method Proves embedding using open book decompositions and spin structures.
result Proves spun embedding for simply connected spin 5-manifolds and 3-dimensional open books.
There is an interpretation of open string field theory in algebraic topology. An interpretation of closed string field theory can be deduced from this open string theory to obtain as well the interpretation of open and closed string field theory combined.
Survey of recent scalar curvature results on open 3-manifolds.
problem None explicitly stated; focuses on recent results.
method Survey and review of existing research.
result Recent findings on scalar curvature in open 3-manifolds.
Extends Morse structures to partial open books for studying contact 3-manifolds.
problem Analyzing contact 3-manifolds with convex boundaries.
method Develops Morse structures on extendable partial open books.
result New tools for studying contact 3-manifolds with convex boundaries.
A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real structure. We call them real open book decompositions. We show that each real open b…