Enhances Vassiliev knot invariants using chord diagrams.
problem Improving Vassiliev knot invariants.
method Generalizes Vassiliev knot invariants to framed chord diagrams.
result Enhances Vassiliev knot invariants.
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48 results for “knot diagram invariants”
New method encodes knots using clasp diagrams for easier study of invariants.
problem Encoding and studying knots and links more efficiently.
method Introducing clasp diagrams and equivalence relations on them.
result Explicit formula for Alexander-Conway polynomial and Vassiliev invariants.
The abstract provides a method to represent knots and generalizes knot invariants.
problem Representing knots and generalizing knot invariants.
method Signed chord diagrams and moves to connect diagrams for the same knot.
result Every knot can be represented by signed chord diagrams and connected by moves.
New diagonal knots found with non-torus structure.
problem Identifying knots with diagonal grid diagrams.
method Analysis of knots represented by diagonal grid diagrams.
result All diagonal knots are positive, and a new non-torus example is found.
Generalizes meander diagrams to virtual knots and introduces new invariants.
problem Classifying and comparing virtual knots.
method Generalization of meander diagrams to virtual knots, introduction of virtual -arc crossing numbers.
result Classes of meander and semimeander diagrams are universal for virtual knots.
The independence system is a knot invariant for alternating knots.
problem Defining and proving the independence system as a knot invariant.
method Defined an independence system for knot diagrams and proved it as an invariant for alternating knots.
result The independence system is a knot invariant for alternating knots.
New formula for knot invariants simplifies calculations and counts.
problem Calculating knot invariants for various diagrams.
method Localized configuration space integral with a new Gauss form.
result Yields arrow diagram expressions and new lower bounds.
Invariants of Knot Diagramsmath.GT
We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.
Delta Diagramsmath.GT
Delta Diagrams show any knot or link can be represented with 3-5 sided regions.
problem Representing knots and links with specific geometric constraints.
method Defined Delta Diagrams and proved any knot/link can be represented.
result Proved any knot/link can be represented by Delta Diagrams.
Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in ; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are defined only for marked vertex diagrams representing knotted orientable surfaces; we e…
Paper provides seven Gauss diagram formulas for degree three long virtual knots.
problem Tackles the complete list of seven distinct Gauss diagram formulas for degree three long virtual knots.
method Gives seven Gauss diagram formulas for degree three long virtual knots and 23 for classical knots.
result Each Gauss diagram formula for degree three long virtual knots is represented as classical knots formulas, supporting Goussarov-Polyak-Viro conjecture.
In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some -moves that only differ in how the three strands that are involved in the move are ordered on the knot. To transform knot diagrams of isotopic knots into each other …
In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Fl…
New knot invariant from Kauffman states and algebraic modules.
problem Developing a new knot invariant related to Alexander polynomial.
method Defining a bigraded knot invariant using Kauffman states and differential graded algebras.
result The invariant's homology corresponds to the Alexander polynomial.
Conjectures closed-form expressions and cyclotomic expansions for knot invariants.
problem Calculating HOMFLY-PT invariants of knots colored by rectangular diagrams.
method Interpolation Macdonald polynomials and cyclotomic expansions.
result Conjectured closed-form expressions and cyclotomic expansions for knot invariants.
Paper discusses a new link invariant from virtual link theory.
problem Developing a new invariant for twisted links.
method Using JKSS invariant of virtual links and double coverings.
result Derived a new invariant for twisted links.
Virtual Knot Groupsmath.GT
New groups defined from knot diagrams, invariant under Reidemeister moves.
problem Classical knot groups are not invariant under all Reidemeister moves.
method Define quotient groups based on knot diagrams, invariant under Reidemeister moves.
result New groups include extended knot groups and are invariant under all Reidemeister moves.
Two complete knot invariants from diagrams, finite or infinite.
problem Classifying knots completely.
method Constructed two invariants from knot diagrams, finite or infinite.
result Finite set reveals knotting number.
The state-sum invariants for knots and knotted surfaces defined from quandle cocycles are described using the Kronecker product between cycles represented by colored knot diagrams and a cocycle of a finite quandle used to color the diagram. Such an interpretation is applied to evaluating the invariants. Algebraic inter…
A combinatorial one-cocycle in a moduli space of knots from the Vassiliev invariant of order 3math.GT
New knot invariant derived from Vassiliev invariant of order 3.
problem Finding new knot invariants using Vassiliev invariants.
method Using Gauss diagram formulas and combinatorial 1-cocycles in moduli space of knots.
result Proved that Vassiliev invariant of order 3 gives a 1-cocycle.
Grid diagrams define invariants for knots in lens spaces.
problem Characterizing knots in lens spaces using grid diagrams.
method Using combinatorial grid diagrams to define invariants.
result Invariants are equivalent to those defined in previous works.
New invariant valued in pictures enhances state-sum invariants.
problem Enhancing state-sum invariants for virtual knots.
method Constructing a picture-valued biquandle bracket.
result Enhances various state-sum invariants for virtual knots.
New algebraic structures biquasiles defined using dual graph diagrams for knot and link invariants.
problem Defining invariants for oriented knots and links.
method Introducing dual graph diagrams and biquasiles, using combinatorial and algebraic approaches.
result Defined new knot and link invariants using biquasiles.
Clarifies how knot homset invariants relate to diagram colorings.
problem Confusion in representing homset elements with similar diagrams.
method Examines and clarifies the relationship between knot homset invariants and diagram colorings, particularly in symmetric cases.
result Identifies conditions under which homset elements can be represented by similar diagrams.
We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite type invariant and show that the restriction of any such invariant of degree n to …
The paper calculates a specific weight system for chord diagrams with a particular graph structure.
problem Calculating a specific weight system for chord diagrams with a complete bipartite graph structure.
method Using a Lie algebra and its weight system, the authors derive a function on chord diagrams.
result The authors compute the weight system for chord diagrams with a complete bipartite graph structure.
Interactive platform for knot invariant computation and identification.
problem Efficient computation and identification of knot invariants.
method Unified web platform using Feynman ribbon diagrams and tensor networks.
result First platform combining construction, evaluation, computation, and identification.
The Witten-Reshetikhin-Turaev invariant of classical link diagrams is generalized to virtual link diagrams. This invariant is unchanged by the framed Reidemeister moves and the Kirby calculus. As a result, it is also an invariant of the 3-manifolds represented by the classical link diagrams. This generalization is used…
Racks do not give us invariants of surface-knots in general. For example, if a surface-knot diagram has branch points (and a rack which we use satisfies some mild condition), then it admits no rack colorings. In this paper, we investigate rack colorings for surface-knot diagrams without branch points and prove that rac…
The paper introduces a new invariant for 2-knots in S^4.
problem Developing a new invariant for 2-knots in S^4.
method Using double-point diagrams and planar geometry to compute the invariant.
result The invariant takes values in Z_2 and can be computed from the double-point diagram.
New approach to electric group for knots and links.
problem No previous publication of electric invariant for knots and links.
method Simple and general approach to electric group for oriented knots and links, using proper colouring of knot diagrams.
result Each homomorphism from the electric group to an arbitrary finite group can be described by a proper colouring of the diagram.
Paper introduces an invariant to distinguish handlebody-knot exteriors.
problem Challenges in distinguishing handlebody-knots with homeomorphic exteriors.
method Defined an invariant (annulus diagram) using characteristic submanifold theory and Koda-Ozawa classification for essential annuli.
result The annulus diagram can differentiate handlebody-knot families.
New method finds knots without low treewidth diagrams.
problem Finding knots without low treewidth diagrams.
method Structural graph theory and knot theory.
result Optimal obstruction for high representativity knots.
Analogous zeta function for twisted Alexander invariants defined.
problem Defining a zeta function for twisted Alexander invariants.
method Modeling random walks on knot diagrams and interpreting Alexander polynomials and Jones polynomials as zeta functions.
result Analogous zeta function expression for twisted Alexander invariants.
In this paper we introduce a representation of a embedded knotted (sometimes Lagrangian) tori in $\BR^4$ called a hypercube diagram, i.e., a 4-dimensional cube diagram. We prove the existence of hypercube homology that is invariant under 4-dimensional cube diagram moves, a homology that is based on knot Floer homology.…
New parities defined on virtual knots linked to crossing indices.
problem Defining parities on virtual knots.
method Connecting parities to invariant cycles on arcs and quasi-indices on crossings.
result New series of parities on virtual knots defined.
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.
Defines knotting probability for spatial arcs.
problem Calculating the probability of knotting in spatial arcs.
method Projection of spatial arcs to planes, defining knotting probability for diagrams.
result Knotting probability defined for every oriented spatial arc.
An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, -genus and -genus of a positive knot are equal. In this paper, we p…
Combinatorial proof for knot Floer homology in branched covers.
problem Proving the invariance of knot Floer homology in branched covers.
method Using a Heegaard diagram and grid diagram for a knot in its cyclic branched cover.
result Combinatorial proof of knot Floer homology invariance over Z.
Develops a calculus for knotted objects, including classical links.
problem Characterizing and classifying knotted objects, including classical links.
method Introduces Arrow presentations and w-tree presentations to encode knot diagrams, uses clasper theory for welded knots.
result Characterizes finite type invariants of welded knots and long knots.
New invariant from knot diagrams helps classify knots.
problem Classifying knots using strong Heegaard invariants.
method Computing from knot Heegaard diagrams.
result Constructs a transverse knot invariant refining existing invariants.
A new knot invariant measures crossings in three orthogonal directions.
problem Defining a new knot invariant for certain knot diagrams.
method Defining the simultaneous crossing number for knots with doubly transvergent diagrams.
result The limit of the ratio of the new invariant to the usual crossing number is at most 8.
Regional knot invariantsmath.GT
A new knot invariant is created using regions and crossings.
problem Creating a new knot invariant for regional and crossing contributions.
method Each planar region and crossing contribute to a generator and relation, respectively, forming a tridle of the link.
result A polynomial invariant can be derived from the presentation matrix of a linear tridle.
In this article we study the differential graded algebra (DGA) invariant associated to Legendrian knots in tight lens spaces. Given a grid number one diagram for a knot in L(p, q), we show how to construct a special Lagrangian diagram suitable for computing the DGA invariant for the Legendrian knot specified by the dia…
New method computes transverse knot invariant using braids.
problem Compute transverse invariant of knots and links.
method Define new combinatorial complex to compute knot Floer homology.
result Identify BRAID invariant of transverse knots and links.
Formula for computing rotation and self-linking numbers in contact surgery diagrams.
problem Computing rotation and self-linking numbers for Legendrian knots and transverse knots in contact surgery diagrams.
method Explicit formula and extension of Ding-Geiges-Stipsicz formula for d3-invariant.
result Explicit formulas for rotation and self-linking numbers in contact (1/n)-surgery diagrams.
This paper describes a polynomial invariant of virtual knots that is defined in terms of an integer labeling of the virtual knot diagram. This labeling is seen to derive from an essentially unique structure of affine flat biquandle for flat virtual diagrams. The invariant is discussed in detail with many examples,inclu…