Knot modules are linked to ribbon 2-knots with geometric constraints.
problem Characterizing knot modules of ribbon 2-knots.
method Showing every Z-torsion free knot module can be realized by a ribbon 2-knot with specific geometric properties. result Knot modules of ribbon 2-knots have geometric constraints with group of geometric dimension at most 2.
Classifies modules of surface-knots in terms of their properties.
problem Characterizing modules of surface-knots in terms of their properties.
method Using homology and covering spaces, the reduced first module is characterized.
result The reduced first module for every genus g is characterized in terms of properties of a finitely generated module.
New knot theory module shows torsion-ness in number theory.
problem Torsion-ness of Selmer modules in Galois representations.
method Introducing adjoint homological Selmer module for SL2-representations of knot groups. result Finitely generated torsion-ness of the new Selmer module.
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.
Combinatorial approach to compute satellite knot invariants using graph theory.
problem Computing knot invariants for satellite knots using bordered Heegaard Floer homology.
method Construct weighted A∞-modules using decorated planar graphs and prove their isomorphism. result Combinatorial proof of A∞ structure relations for the constructed modules. Enhances knot and link invariant using tribracket modules.
problem Counting invariants of oriented knots and links.
method Introduces tribracket modules and uses them to enhance the tribracket counting invariant.
result Shows the enhancement is proper and provides examples.
Paper computes skein modules of 3-manifolds using braids.
problem Computing skein modules of 3-manifolds.
method Using braids and knot algebras.
result Braid approach to HOMFLYPT and Kauffman bracket skein modules of specific 3-manifolds.
Cochran defined the nth-order integral Alexander module of a knot in the three sphere as the first homology group of the knot's (n+1)th-iterated abelian cover. The case n=0 gives the classical Alexander module (and polynomial). After a localization, one can get a finitely presented module over a principal ideal domain,…
The classical abelian invariants of a knot are the Alexander module, which is the first homology group of the the unique infinite cyclic covering space of S^3-K, considered as a module over the (commutative) Laurent polynomial ring, and the Blanchfield linking pairing defined on this module. From the perspective of the…
In this article, we give a classification of Alexander modules of null-homologous knots in rational homology spheres. We characterize these modules A equipped with their Blanchfield forms φ, and the modules A such that there is a unique isomorphism class of (A,φ), and we prove that for the other modules A, there ar…
We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non trivial by calculating explicit examples. We define welded virtual graphs and con…
Corrects a paper on Alexander modules and answers a related question.
problem Peripheral elements in reduced Alexander modules
method Analyzes and corrects a paper on Alexander modules
result Answers a question and corrects a minor error in the original paper
Using computational techniques we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces L(p,q). For these knots we calculate the second and third skein module and establish which prime knots in the solid torus are amphichiral. Most knots are distinguished by the skein mod…
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 Z[q±21] for specific knots. result Provides a negative answer to Problem 1.92 in Kirby's list.
We construct examples of knots that have isomorphic nth-order Alexander modules, but non-isomorphic nth-order linking forms, showing that the linking forms provide more information than the modules alone. This generalizes work of Trotter, who found examples of knots that have isomorphic classical Alexander modules, but…
GridPyM handles grid diagrams for knot theory.
problem Handling grid diagrams for knot theory.
method Generates and simplifies grids, models local transformations.
result Models local transformations between grid diagrams.
A criterion ensures double sliceness for certain knots and satellite knots.
problem Ensuring double sliceness for specific types of knots and satellite knots.
method Using noncommutative Alexander modules and derived subgroup conditions.
result A condition for doubly slice knots and satellite knots is established.
Model proteins with bonds using Kauffman bracket skein module.
problem Modeling proteins with bonds for structural analysis.
method Extend Kauffman bracket polynomial to bonded knots.
result Infinite generation and torsion-freeness of the bonded skein module.
An invariant for knots with colored bonds respects HOMFLYPT relation.
problem Modeling and distinguishing knots with colored bonds.
method Introducing a HOMFLYPT skein module for colored bonded knots.
result The non-rigid version of the module provides information about knottedness of bonds.
We compute the Kauffman skein module of the complement of torus knots in S^3. Precisely, we show that these modules are isomorphic to the algebra of Sl(2,C)-characters tensored with the ring of Laurent polynomials.
Proves module structure on odd Khovanov homology and applies to ribbon 2-knots.
problem Proving module structure on odd Khovanov homology.
method Geometric description of module structure and link cobordism.
result Combinatorial proof of odd invariant for ribbon 2-knots.
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.
Study the algebraic action of torus on knot complement's skein module.
problem Understand the algebraic structure of knot complements and boundary tori.
method Analyze the Kauffman bracket skein algebra and module of the 3-twist knot complement.
result Determine the action of Kauffman bracket skein algebra on module of 3-twist knot complement.
We study the twisted knot module for the universal deformation of an SL2-representation of a knot group, and introduce an associated L-function, which may be seen as an analogue of the algebraic p-adic L-function associated to the Selmer module for the universal deformation of a Galois representation. We…
Quantum theory constructs a group and skein module for knot complements.
problem Understanding the fundamental group of knot complements using quantum methods.
method Using bottom tangles, the universal space of quantum representations is constructed, then factored by the skein relation to get the skein module.
result Derives recurrence relation for the colored Jones polynomial, known as Aq polynomial. 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. Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander module…
We define and study a bigraded knot invariant whose Euler characteristic is the Alexander polynomial, closely connected to knot Floer homology. The invariant is the homology of a chain complex whose generators correspond to Kauffman states for a knot diagram. The definition uses decompositions of knot diagrams: to a co…
Unified invariant of knots derived from Verma modules.
problem Constructing a unified invariant of knots from quantum sl2.
method Braid groups' action on tensors of Verma modules.
result Unified invariant interpolates colored Jones and ADO polynomials.
In the bordered Floer theory, gluing thickened torus of positive meridional Dehn twist to the boundary of a knot complement result in the knot complement of increased framing. For a fixed knot K, we construct a direct system of positively framed knot complements and study the direct limit. We also study the morphism sp…
This paper is a brief overview of some of our recent results in collaboration with other authors. The cocycle invariants of classical knots and knotted surfaces are summarized, and some applications are presented.
Survey of knot invariants in lens spaces.
problem Comparing knot invariants in lens spaces.
method Summarized various invariants and compared their distinguishing power.
result Invariants generalize classical knot polynomials and can distinguish difficult cases.
Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.
problem Understanding HOMFLYPT polynomials and their geometric origins.
method Defining worldsheet skein module and D-module, considering skein valued open curve counts.
result Worldsheet skein D-module for Hopf link conormal is generated by three operator polynomials.
New findings on hyperbolic 3-manifolds with finite skein modules.
problem Finite dimensional skein modules of hyperbolic 3-manifolds.
method Introduced and proved stability property for Dehn-filling of knots.
result Almost all Dehn-fillings of two-bridge knots have finite skein modules.
We prove that if M is a CW-complex and M1 is its 1-skeleton then the crossed module Π2(M,M1) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π2(D2,S1). From this it follows that, if G is a finite crossed module and M is finite, then th…
Study connects knot contact homology to Chern-Simons theory's large N limit.
problem Relate knot contact homology to Chern-Simons theory's large N limit.
method Prove conjecture linking augmentation varieties to Chern-Simons theory's large N limit; characterize HOMFLYPT difference module.
result Classical limit of HOMFLYPT difference module equals degree 0 abelianized knot contact homology.
We introduce the Alexander-Beck module of a knot as a canonical refinement of the classical Alexander module, and we prove that this new invariant is an unknot-detector.
Study on the dimension of skein modules of Dehn fillings for specific knots.
problem Determining the dimension of skein modules for Dehn fillings of knots.
method Analyzing Kauffman bracket skein modules and using character varieties.
result Dimension of skein modules for almost all primitive roots of unity and slopes.
Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum g…
New quantum knot invariants derived from Verma modules.
problem Constructing universal quantum knot invariants from Verma modules.
method Defining level N universal invariants from finite quotients of Verma modules over quotient rings.
result Maximal universal invariants for prime N, interpolating Jones and ADO polynomials.
We compute the Kauffman bracket skein module of the complement of a twist knot, finding that it is free and infinite dimensional. The basis consists of cables of a two-component link, one component of which is a meridian of the knot. The cabling of the meridian can be arbitrarily large while the cabling of the other co…
Python tool calculates cobordism maps in Khovanov homology.
problem Computing cobordism maps on Khovanov homology.
method Developed a Python module to calculate these maps.
result Computed cobordism maps for all incompressible Seifert surfaces of prime knots up to 10 crossings.
Study satellite knots and their quandles related to incompressible tori.
problem Understanding quandles of satellite knots and their components.
method Algebraic approach to augmented fundamental quandles, presentations of fundamental quandles, and analysis of Alexander modules.
result Relationships between satellite knots, companion and pattern knots, and their fundamental quandles.
We construct the first combinatorial 1-cocycle with values in the Z[x,x−1]-module of isotopy classes of singular long knots in 3-space with a signed planar double point, and which represents a non trivial cohomology class in the topological moduli space of long knots. It can be interpreted as an invaria…
Study corrects previous work on knot Floer homology of certain pretzel knots.
problem Computing the Knot Floer Homology of specific pretzel knots.
method Applied peculiar modules theory for Floer homology of 4-ended tangles, using immersed curve interpretation.
result Corrected the rank of Knot Floer Homology for some pretzel knots.
Khovanov homology distinguishes exotic 4-manifolds.
problem Existence of exotic compact orientable 4-manifolds.
method Khovanov-Rozansky gl2 skein lasagna module and related invariants. result First analysis-free proof of exotic 4-manifolds.
New knot invariant from braided Hopf algebra.
problem Developing a new knot invariant.
method Non-commutative generalization of knot groups using braided Hopf algebra.
result New quantum character variety as an alternative to skein module.
Positive knots are minimal in a specific knot ordering.
problem Understanding the minimal structure of positive knots in a specific knot ordering.
method Analyzing ribbon concordance and using properties of rational Alexander modules.
result Positive knots are minimal in the ribbon concordance ordering.