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.
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.
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.
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.
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.
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.
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.
Study links and quivers, proving polynomial equality conjecture.
problem Link and quiver invariants and their relations.
method Cluster algebra invariants, point count polynomials, skein relations.
result Equality conjecture between plabic graph link polynomial and quiver point count polynomial proved for specific cases.
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.
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.
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.
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.
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.
The paper generalizes knot invariants and their connections to quivers and ideals.
problem Understanding knot complements and their invariants.
method Generalizing FK invariants, knots-quivers correspondence, and A-polynomials; associating FK to branch of A-polynomial; quiver generating series; R-matrices; quantum a-deformed A-polynomial; 3d-5d theory. result Explicit expressions for FK invariants and their quiver representations for several simple knots. 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.
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.
Enhances knot and link invariants using quandle modules.
problem Distinguishing knots and links using polynomial invariants.
method Integrates quandle modules into the quandle coloring quiver.
result The enhanced invariant distinguishes knots and links.
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
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.
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.
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.
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…
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.
New invariants defined for knots and links using quandle representations.
problem Defining new invariants for knots and links.
method Defined a family of quiver representations associated to finite quandles, abelian groups, and quandle 2-cocycles.
result Computed four new polynomial invariants for knots and links.
Ordinarily, quiver varieties are constructed as moduli spaces of quiver representations in the category of vector spaces. It is also natural to consider quiver representations in a richer category, namely that of vector bundles on some complex variety equipped with a fixed sheaf that twists the morphisms. Representatio…
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.
It is known that there are only finitely many knots with super bridge index 3. Jin and Jeon have provided a list of possible such candidates. However, they conjectured that the only knots with super bridge index 3 are trefoil and the figure eight knot. In this paper, we prove that the 52 knot and the 62 knot are …
The recently conjectured knots-quivers correspondence relates gauge theoretic invariants of a knot K in the 3-sphere to representation theory of a quiver QK associated to the knot. In this paper we provide geometric and physical contexts for this conjecture within the framework of the large N duality of Ooguri…
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.
Community detection is a central problem of network data analysis. Given a network, the goal of community detection is to partition the network nodes into a small number of clusters, which could often help reveal interesting structures. The present paper studies community detection in Degree-Corrected Block Models (DCB…
We use computer algebra to demonstrate the existence of a multilinear polynomial identity of degree 8 satisfied by the bilinear operation in every Lie-Yamaguti algebra. This identity is a consequence of the defining identities for Lie-Yamaguti algebras, but is not a consequence of anticommutativity. We give an explicit…
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.
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.
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.
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.
The paper constructs tilting modules for knots using algebraic structures.
problem Understanding the algebraic structure of knot invariants.
method Constructing modules over Jacobian algebras associated with knots.
result The constructed modules M are rigid and τ-rigid, and their endomorphism algebra is isomorphic to the Jacobian algebra. 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.
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…
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.
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.
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.