Study of quandle coloring quivers with dihedral quandles.
problem Link invariants and their enhancements using quandles.
method Introduced shadow quandle coloring quivers and cocycle quivers, studied equivalence with quandle coloring numbers and shadow quandle cocycle invariants.
result Equivalence of quandle coloring quivers with quandle coloring numbers and shadow quandle cocycle quivers with shadow quandle cocycle invariants for specific dihedral quandles.
The study characterizes torus links' coloring quivers using dihedral quandles.
problem Characterizing the structure of coloring quivers for torus links.
method Exhaustively determining all possible numbers of colorings and their interconnections.
result The quiver structure varies based on the number of colorings.
Study on quandle coloring quivers for (p, 2)-torus knots and links.
problem Understanding quandle colorings of (p, 2)-torus knots and links.
method Introduced quandle coloring quivers and studied them for dihedral quandles.
result Characterized quandle coloring quivers for (p, 2)-torus knots and links.
In-degree quiver polynomials for surface-links computed.
problem Computing in-degree quiver polynomials for surface-links.
method Defined using a quandle and set of endomorphisms, computed for surface-links with ch-index up to 10.
result Example computations for surface-links with ch-index up to 10.
Enhances psyquandle invariants for singular and pseudoknots.
problem Counting invariants for singular knots and pseudoknots.
method Uses quivers to extend in-degree polynomial invariants.
result Obtains biquandle coloring quivers and in-degree polynomial invariants.
New polynomial invariants from quandle action quivers.
problem Classical and virtual knot and link invariants.
method Categorification of quandle counting invariant using quandle action quivers.
result Quandle action polynomials as decategorifications.
Enhanced invariant for linkoids using quivers.
problem Counting invariants for linkoids.
method Use of quivers to generalize in-degree polynomial invariant.
result Introduced in-degree quiver polynomial matrix as a new invariant.
New polynomial invariants for knots and links from quandle coloring quiver decategorification.
problem Defining new polynomial invariants for knots and links.
method Decategorification of the quandle coloring quiver to create polynomial invariants.
result The invariants are not determined by the quandle counting invariant.
We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most literal sense, i.e., giving the set of quandle colorings the structure of a small…
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.
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.
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.
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.
New geometric proof for rational tangles links-quivers correspondence.
problem Recovering symmetric/antisymmetric colored HOMFLY-PT polynomials from a quiver.
method Geometric approach using winding numbers in punctured plane and its second configuration space.
result Explicit description of quivers for rational tangles.
The abstract conjectures a link between knot homologies and quiver partition functions.
problem Understanding the relationship between knot homologies and quiver partition functions.
method Interpreting quiver nodes as holomorphic curves with boundary on the knot conormal, and studying recursion relations.
result Generalized quiver partition functions are related to knot homologies.
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
New polynomial invariants derived from birack and switch structures.
problem Polynomial invariants of braids.
method Switch structures, birack colorings, quiver-valued invariants.
result New polynomial invariants of braids.
We incorporate quandle cocycle information into the quandle coloring quivers we defined in arXiv:1807.10465 to define weighted directed graph-valued invariants of oriented links we call \textit{quandle cocycle quivers}. This construction turns the quandle cocycle invariant into a small category, yielding a categorifica…
Study bridges between biquandles, quivers, and virtual link bridge numbers.
problem Understanding the gap between different formulations of bridge numbers for virtual links.
method Investigates connections between biquandle colorings, quiver enhancements, and bridge numbers.
result Shows existence of virtual links with specific bridge numbers and constructs families distinguishing quiver invariants.
Geometrically describes the linear and quadratic forms for rational links.
problem Predicting generating functions for colored HOMFLY-PT polynomials of rational links.
method Direct geometric description of linear and quadratic forms in terms of configuration spaces.
result Direct geometric description of forms for rational links.
We enhance the quandle coloring quiver invariant of oriented knots and links with quandle modules. This results in a two-variable polynomial invariant with specializes to the previous quandle module polynomial invariant as well as to the quandle counting invariant. We provide example computations to show that the enhan…
New invariants defined for framed knots and links.
problem Defining invariants for framed knots and links.
method Introducing birack brackets and categorifying their multiset.
result Quiver-valued invariant defined for framed knots and links.
Study lattice paths from twist knots and double twist knots.
problem Understanding combinatorics of twist knots and double twist knots.
method Analyzing quiver generating series of HOMFLY-PT polynomial limits.
result Lattice path models for twist knots and double twist knots.
This work reconstructs knot invariants from Alexander polynomials, proving consistency with known theorems.
problem Reconstructing knot invariants from Alexander polynomials.
method Quantization, deformation, and rewriting of Alexander polynomials.
result Derives new formulae for colored superpolynomials and proves consistency with Melvin-Morton-Rozansky theorem.
The thesis examines the relationship between three-manifold invariants and knot theory, finding equalities and patterns.
problem Examining the relationship between three-manifold invariants and knot theory.
method Analytic continuation and quiver representation theory.
result Found equalities and patterns in knot theory and quiver representation.
Classifies singularities in quiver varieties for specific Dynkin quivers.
problem Classifying singularities in quiver varieties.
method Classifies singularities using minimal imaginary roots and extended Dynkin quivers.
result Constructs hyper-Kähler cobordisms between binary polyhedral spaces.
The paper studies the geometry of Nakajima quiver varieties and their decompositions.
problem Understanding the geometry of Nakajima quiver varieties and their decompositions.
method Investigates the Białynicki--Birula decomposition of Nakajima quiver varieties, describing fixed points in terms of representations with relations of auxiliary quivers.
result Computes the motivic decomposition of Nakajima quiver varieties in terms of quiver-chain moduli spaces.
This review connects knot invariants to quiver representations.
problem Relating knot invariants to quiver representations.
method Relates symmetric quivers and their partition functions to quantum invariants of knots.
result Establishes a correspondence between knot invariants and quiver representations.
Theory of smooth relative connections on quiver bundles developed.
problem Existence of smooth relative connections over quiver bundles.
method Developed a theory over RQ on smooth twisted quiver bundles, provided obstructions and necessary/sufficient conditions. result Established a necessary and sufficient condition for the existence of smooth relative connections on tree-type quiver bundles.
We define and calculate the HOMFLY polynomial for a specific type of quiver.
problem Calculating the HOMFLY polynomial for forest quivers.
method Recursive definition and closed-form expression for forest quivers.
result Closed-form expression for the HOMFLY polynomial of a forest quiver.
Symmetrizes 4d and 3d BPS quivers for Argyres-Douglas theories.
problem Understanding the relationship between 4d and 3d BPS quivers.
method Analyzes geometric backgrounds and uses skein modules to derive quiver partition functions.
result Proves isomorphism between 4d wall-crossing and unlinking of symmetric quivers.
Let G be a Lie group and Q a quiver with relations. In this paper, we define G-valued representations of Q which directly generalize G-valued representations of finitely generated groups. Although as G-spaces, the G-valued quiver representations are more general than G-valued representations of finitely generated group…
Knot invariants and quiver stability linked through full twists.
problem Relating knot invariants to quiver stability under twists.
method Using HOMFLY-PT skein relations and linking/unlinking operations on symmetric quivers.
result Full twists on knots correspond to unlinking or linking of augmented symmetric quivers, confirming stable growth in both.
Introduces tensor product for quiver representations and applies to stable bundles and character varieties.
problem Stability and classification of quiver bundles and their subvarieties.
method Definition of tensor product for quiver representations and application to stability and character varieties.
result Tensor products of polystable quiver bundles are polystable and provide insights into character varieties.
Study of conformal limits in Nakajima quiver varieties.
problem Understanding the conformal limits of Nakajima quiver varieties.
method Defined and studied a conformal limit construction for Nakajima quiver varieties, proving it is a limit of a one-parameter family and gives a biholomorphic map.
result Proved the conformal limit is a biholomorphic map between Lagrangian submanifolds of different quiver varieties.
The relation between open topological strings and representation theory of symmetric quivers is explored beyond the original setting of the knot-quiver correspondence. Multiple cover generalizations of the skein relation for boundaries of holomorphic disks on a Lagrangian brane are observed to generate dual quiver desc…
Introduces Nakajima bundles on algebraic curves, generalizing quiver representations and bundles.
problem Generalizing quiver representations and bundles on algebraic curves.
method Assigns complex vector bundles and sections/connections to nodes and edges of a quiver, using gauge-theoretic characterizations.
result Proves Hitchin-Kobayashi correspondence between stable quiver bundles and Nakajima bundle representations.
New method constructs nilpotent Lie algebras from quivers.
problem Constructing nilpotent Lie algebras from quivers.
method Using paths within quivers to construct nilpotent Lie algebras.
result Constructs a broad family of Ricci soliton nilmanifolds.
Forbidden moves categorify fused links into quivers.
problem Categorify fused links to invariant of links.
method Use forbidden moves to categorify fused links.
result Obtain three polynomial invariants of links.
We argue how to identify supersymmetric quiver quantum mechanics description of BPS states, which arise in string theory in brane systems representing knots. This leads to a surprising relation between knots and quivers: to a given knot we associate a quiver, so that various types of knot invariants are expressed in te…
Neural networks are mathematically represented via quiver representations.
problem Understanding how neural networks process data and create representations.
method Representing neural networks as quiver representations with activation functions.
result Neural networks' computations can be studied algebraically and geometrically.
New Lie algebras from quivers lead to rigid Ricci solitons.
problem Constructing Lie algebras from quivers to study geometric structures.
method Using finite quivers without cycles to construct solvable Lie algebras and proving their geometric properties.
result Simply-connected Lie groups corresponding to these Lie algebras admit left-invariant Ricci solitons, and when quivers are oriented multi-trees, these groups are rigid.
Paper develops Morse-theoretic approach to quiver varieties convolution.
problem Convolution in quiver varieties via Morse theory.
method Morse complex and cup product on smooth space of representations.
result Topological information encoded in cup product of Morse complex.
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.
We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every …
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.
The paper connects quivers to knot complements and studies their BPS states and 3d N=2 theories.
problem Understanding the relationship between quivers and knot complements.
method Assigning quivers to knot complements and exploring their physical interpretation.
result Proposed a physical interpretation of quivers in terms of BPS states and 3d N=2 theories.
Survey of knot polynomials and their categorification, including quiver-knot correspondence.
problem Understanding the relationship between knot polynomials and quivers.
method Overview of classical knot polynomials, physical and geometric insights, and 3d N=2 theory analysis. result Exploration of the LMOV invariants and their connection to BPS states.