Explains biquandle brackets and quivers for a topology talk.
problem None explicitly stated; focuses on background information.
method Review of biquandle concepts and related structures.
result Clarifies understanding of biquandle bracket quivers.
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48 results for “topological biquandle”
A new topological biquandle for links defined.
problem Defining a topological biquandle for links.
method Topologically defined biquandle construction similar to fundamental quandle.
result Presentation of topological biquandle and its relation to fundamental biquandle.
New topological biquandles created using skew braces.
problem Creating nontrivial topological biquandles.
method Using the concept of skew braces.
result Constructs nontrivial examples of topological biquandles.
Enhances quandle and biquandle colorings using biquandle structures.
problem Improving quandle and biquandle colorings.
method Investigates biquandle structures and biquandle homomorphisms.
result Describes biquandle structure of the Hom-biquandle and relates Hom-quandles to Hom-biquandles.
Constructing biquandlesmath.GR
Defines biquandle structures and relates automorphism groups of biquandles and quandles.
problem Understanding the structure and automorphism groups of biquandles.
method Define biquandle structures, determine isomorphisms, and compute automorphism groups.
result Establishes a relationship between biquandle and quandle automorphism groups.
Shadow biquandles and local biquandles have similar homology and invariants.
problem Comparing homology and invariants of shadow biquandles and local biquandles.
method Defined local biquandle structure on a shadow biquandle and showed isomorphic (co)homology groups and invariant equivalence.
result Homology and invariants of shadow biquandles and local biquandles are equivalent.
New biquandle bracket invariants are linked to biquandle 2-cocycles.
problem Quantum enhancements and biquandle colored links.
method Proving biquandle bracket invariants are pointwise products of other invariants and biquandle 2-cocycles.
result New biquandle bracket invariants are equivalent to the Jones polynomial on knots.
Biquandle Bracket Quiversmath.GT
New categorifications of biquandle brackets defined.
problem Categorify biquandle invariants of knots and links.
method Define biquandle bracket quivers to enhance biquandle counting invariants.
result Provides an infinite family of categorifications of the Jones polynomial.
This paper connects virtual biquandles to biquandles for virtual link colorings.
problem Extending invariants from biquandles to virtual biquandles.
method Establishing equivalence between two representations of virtual braid groups and introducing new labeling rules.
result The number of colorings of a virtual link by virtual biquandles can be recovered from colorings by biquandles.
New knot invariants derived from biquandle quivers.
problem Enhancing knot invariants for virtual and classical knots.
method Categorification using biquandle coloring quivers.
result New infinite families of polynomial invariants.
Paper discusses biquandle cohomology and invariants for surface-links.
problem Computing invariants for surface-links using biquandle theory.
method Developed (co)homology theory of biquandles, introduced new invariants using broken surface diagrams and marked graph diagrams.
result Constructs new invariants for surface-links using biquandle theory.
Researchers attempt to categorify biquandle brackets using Khovanov homology methods.
problem Categorify biquandle brackets using Khovanov homology methods.
method Outline a Khovanov homology-style construction for biquandle brackets.
result A canonical biquandle 2-cocycle is defined, but not a true categorification of biquandle brackets.
Biquandle Power Bracketsmath.GT
New invariants for link analysis include biquandle power brackets.
problem Analyzing oriented links with new invariants.
method Introducing biquandle power brackets as an infinite family of link invariants.
result Biquandle power brackets encompass classical and previous invariants.
Finite Type Enhancementsmath.GT
We enhance the biquandle counting invariant using elements of truncated biquandle-labeled Polyak algebras. These finite type enhancements reduce to the finite type enhancements defined by Goussarov, Polyak and Viro for the trivial biquandle of one element and determine (but are not determined by) the biquandle counting…
Parity Biquandlesmath.GT
We use crossing parity to construct a generalization of biquandles for virtual knots which we call Parity Biquandles. These structures include all biquandles as a standard example referred to as the even parity biquandle. Additionally, we find all Parity Biquandles arising from the Alexander Biquandle and Quaternionic …
Innovates a new algebraic structure for handlebody-links.
problem Defining invariants for handlebody-links.
method Introduces multiple conjugation biquandle and extends biquandle operations.
result Establishes a universal algebra for handlebody-links coloring.
Enhances knotoid counting invariants using biquandle brackets.
problem Counting and distinguishing knots and links.
method Defines biquandle brackets with skein relations and coefficients, enhancing biquandle counting matrix invariant.
result New invariants are stronger than previous ones.
New invariants for surface-links using biquandle modules.
problem Defining invariants for surface-links that are not determined by Alexander ideals.
method Enhancing biquandle counting invariant with biquandle modules.
result New invariants not determined by Alexander ideals.
This paper establishes a correspondence between biquandle and quandle colorings for classical and surface links.
problem Finding refined invariants for classical and surface links using biquandles.
method Explicit one-to-one correspondence between biquandle colorings and quandle colorings.
result Biquandle homotopy invariants and quandle homotopy invariants are equivalent.
Kaestner Bracketsmath.GT
A new quantum invariant for virtual knots and links.
problem Quantum invariants for virtual knots and links.
method Generalization of biquandle brackets to parity biquandles.
result The new invariant is stronger than classical biquandle brackets for virtual knots.
New algebraic structures help categorify link invariants.
problem Classifying and distinguishing links and virtual links.
method Introducing mc-biquandles and categorifying homsets.
result New link invariants defined via mc-biquandle coloring quivers.
Define quiver representation-valued invariants for classical and virtual knots
problem Define quiver representation-valued invariants for classical and virtual knots
method Define an infinite family of quiver representation-valued invariants of classical and virtual knots associated to a choice of data vector consisting of a biquandle, abelian group, set of biquandle arrows weights with values in the abelian group, coefficient ring and set of biquandle endomorphisms.
result Extract four new polynomial invariants as decategorifications
We describe a way of representing finite biquandles with n elements as 2n x 2n block matrices. Any finite biquandle defines an invariant of virtual knots through counting homomorphisms. The counting invariants of non-quandle biquandles can reveal information not present in the knot quandle, such as the non-triviality o…
Enhances biquandle invariants for knotoids, detecting mirror images.
problem Detecting mirror images of knotoids using biquandle colorings.
method Constructs new invariants by enhancing the biquandle counting invariant and generalizing Niebrzydowski's longitude invariant.
result Biquandle enhancements detect mirror images of knotoids not distinguishable by the counting invariant.
New method calculates knot and link biquandle brackets using trace diagrams.
problem Computing biquandle brackets of knots and links efficiently.
method Using trace diagrams to compute biquandle brackets of oriented knots and links.
result Identified algebraic conditions for strand moves and stop conditions.
New knot invariants from biquandle arrow weights.
problem Creating nontrivial knot invariants.
method Introducing biquandle arrow weights for knot diagrams.
result Enhancements are nontrivial and not determined by homset cardinality.
The paper introduces biquandle (co)homology and handles invariants for handlebody-links.
problem Developing invariants for handlebody-links using biquandle (co)homology.
method Introducing biquandle (co)homology and constructing invariants using 2-cocycles.
result Constructs invariants for S^1-oriented handlebody-links using 2-cocycles.
Biquandle Virtual Bracketsmath.GT
New quantum invariants for knots and links using biquandle virtual brackets.
problem Quantum enhancements of biquandle counting invariant.
method Defined using skein invariants of biquandle colored diagrams with virtual crossings.
result Includes classical and new invariants, forming an infinite family.
We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY…
The paper explores constructions and symmetries of biquandles from groups and quandles.
problem Understanding symmetries and constructions of biquandles.
method Natural constructions of biquandles from groups and quandles, including holomorph biquandles.
result Determination of automorphism groups of biquandles in terms of associated quandles.
New examples show non-trivial parity-biquandle bracket.
problem Constructing non-trivial parity-biquandle bracket examples.
method Slightly changed notation and constructed examples of knots and links.
result Minimality theorem: graphs appear as link invariants.
New polynomial invariants for knots and links.
problem Defining new invariants for knot theory.
method Infinite family of quiver representations.
result Infinite family of two-variable polynomial invariants.
This paper develops a new homology theory for biquandles and discusses geometric realizations.
problem Constructing knot invariants using set-theoretic Yang-Baxter equation.
method Developed a normalized (co)homology theory for biquandles and geometrically realized them.
result Geometric realization of biquandles has finitely generated second homotopy group for finite biquandles.
The paper establishes a correspondence between quandle and biquandle colorings.
problem Understanding the relationship between quandle and biquandle colorings.
method Defining a functor and showing a one-to-one correspondence between colorings.
result The set of Alexander biquandles colorings is isomorphic to that of Alexander quandles colorings.
We introduce several algebraic structures related to handlebody-knots, including -families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in -oriented spatial trivalent graph diagrams representing -oriented handlebody-k…
Local biquandles link link coloring to tribracket theory.
problem Link coloring and tribracket theory.
method Introduced local biquandles and defined their cohomology.
result Local biquandle cohomology is isomorphic to Niebrzydowski's tribracket cohomology.
Biquandles for Virtual Knotsmath.GT
This paper studies an algebraic invariant of virtual knots called the biquandle. The biquandle generalizes the fundamental group and the quandle of virtual knots. The approach taken in this paper to the biquandle emphasizes understanding its structure in terms of compositions of morphisms, where elementary morphisms ar…
Lower bounds for handlebody-knots using Alexander biquandle colorings.
problem Calculating Gordian distance and unknotting number for handlebody-knots.
method Using Alexander biquandle colorings to construct and calculate handlebody-knots.
result Constructed handlebody-knots with specific distance and unknotting numbers.
On bilinear biquandlesmath.QA
We define a type of biquandle which is a generalization of symplectic quandles. We use the extra structure of these bilinear biquandles to define new knot and link invariants and give some examples.
Quasi-trivial Quandles and Biquandles, Cocycle Enhancements and Link-Homotopy of Pretzel linksmath.GT
Study on link-homotopy of pretzel links using quasi-trivial quandles and biquandles.
problem Link-homotopy of pretzel links.
method Investigation of quasi-trivial quandles and biquandles, cocycle enhancements.
result Necessary and sufficient condition for pretzel links to be trivial under link-homotopy.
New invariants for virtual knots and links defined via quiver representations.
problem Defining new invariants for virtual knots and links.
method Quiver representations associated to virtual biquandles and rings.
result New polynomial invariants for virtual knots and links.
Unified invariant for immersed surface-links using biquandle cocycles.
problem Invariants for immersed surface-links in 4-space.
method Biquandle cohomology, Roseman moves, singular biquandle 3-cocycles.
result Invariance of state-sum invariant under all generating moves, including singular move (h).
New knot invariants using biquandle fares.
problem Creating new knot invariants.
method Using biquandle fares to map paths in diagrams to abelian groups.
result Proper enhancements of invariants for oriented knots and links.
We introduce two kinds of structures, called v-structures and t-structures, on biquandles. These structures are used for colorings of diagrams of virtual links and twisted links such that the numbers of colorings are invariants. Given a biquandle or a quandle, we give a method of constructing a biquandle with these str…
This paper solves Yang-Baxter cohomology for cyclic biquandles.
problem Yang-Baxter cohomology of cyclic biquandles.
method Completely determined free parts and computed torsion subgroups of homology groups.
result Upper bounds for torsion orders in higher dimensional homology groups.
The paper calculates minimum Dehn colors for knots using symmetric local biquandle cocycles.
problem Determining the minimum number of Dehn colors for knots.
method Using symmetric local biquandle cocycle invariants to evaluate minimum Dehn colors.
result There exist knots distinguished by minimum numbers of Dehn colors.
The paper defines biquandles for groups and constructs a homomorphism.
problem Describing dynamical systems in configuration systems.
method Defined biquandles on groups and constructed a homomorphism.
result Homomorphism from surface singular braid monoid to .
It is known that the number of biquandle colorings of a long virtual knot diagram, with a fixed color of the initial arc, is a knot invariant. In this paper we describe a more subtle invariant: a family of biquandle endomorphisms obtained from the set of colorings and longitudinal information.