Non-classical virtual knots may have non-isomorphic upper and lower quandles. We exploit this property to define the quandle difference invariant, which can detect non-classicality by comparing the numbers of homomorphisms into a finite quandle from a virtual knot's upper and lower quandles. The invariants for small-or…
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Kaestner Bracketsmath.GT
We introduce \textit{Kaestner brackets}, a generalization of biquandle brackets to the case of parity biquandles. This infinite set of quantum enhancements of the biquandle counting invariant for oriented virtual knots and links includes the classical quantum invariants, the quandle and biquandle -cocycle invariants…
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 …
We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virt…
Biquandles are generalizations of quandles. As well as quandles, biquandles give us many invariants for oriented classical/virtual/surface links. Some invariants derived from biquandles are known to be stronger than those from quandles for virtual links. However, we have not found an essentially refined invariant for c…
We present a construction of invariants for links using an isomorphism theorem for affine Yokonuma--Hecke algebras. The isomorphism relates affine Yokonuma--Hecke algebras with usual affine Hecke algebras. We use it to construct a large class of Markov traces on affine Yokonuma--Hecke algebras, and in turn, to produce …
The paper introduces a new filtration for knot invariants and proves the existence of nontrivial knots.
problem The existence of nontrivial knots with specific invariant properties.
method Definition of F-order and n-triviality via virtualization and forbidden moves.
result Existence of infinitely many nontrivial classical knots and a nontrivial virtual knot with specific invariant properties.
New probabilistic invariants bound classical topological complexity and category.
problem Bounding classical topological complexity and category.
method Developed probabilistic variants of one-category and diagonal topological complexity.
result Identified new invariants with distributional category and complexity on Eilenberg-Mac Lane spaces.
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.
We look into computational aspects of two classical knot invariants. We look for ways of simplifying the computation of the coloring invariant and of the Alexander module. We support our ideas with explicit computations on pretzel knots.
Combines combinatorial method to extend Milnor invariants to welded links.
problem Extending Milnor invariants to welded links.
method Combinatorial approach.
result Invariance of extended Milnor invariants for welded links.
New Invariants of Knotoidsmath.GT
In this paper we construct new invariants of knotoids including the odd writhe, the parity bracket polynomial, the affine index polynomial and the arrow polynomial, and give an introduction to the theory of virtual knotoids. The invariants in this paper are defined for classical knotoids in analogy to corresponding inv…
We extend several classical invariants of links in the 3-sphere to links in so-called quasi-cylinders. These invariants include the linking number, the Seifert form, the Alexander module, the Alexander-Conway polynomial and the Murasugi-Tristram-Levine signatures.
New invariant distinguishes singular knots and links.
problem Classifying singular knots and links.
method Using oriented singquandles and weight functions at crossings.
result Distinguishes singular granny knot from singular square knot.
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…
A new knot invariant uses permutations to extend Jones polynomials.
problem Extending Jones polynomials to classical and virtual knots and links.
method Colorings by permutations of a finite set to define new knot invariants.
result Established properties and computed polynomials for small cases.
Study probabilistic category and complexity bounds, comparing with classical invariants.
problem Bounding classical category and complexity in probabilistic settings.
method Probabilistic Lusternik-Schnirelmann category and topological complexity computations.
result Established a universal upper bound in finite cases, contrasting with classical invariants.
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 constructs a HOMFLYPT-type invariant for pseudo links.
problem Inability to construct polynomial invariants for pseudo links using Hecke algebra techniques.
method Using a resolution homomorphism and pseudo Hecke algebra of type \(A\), the paper constructs a HOMFLYPT-type invariant for oriented pseudo links.
result The constructed invariant satisfies a natural pseudo skein relation and admits a state-sum formulation.
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.
This work generalizes Hamiltonian mechanics using closed differential forms.
problem Hidden invariants in classical Hamiltonian mechanics.
method Establishes a novel correspondence between generalized Hamiltonian mechanics and multisymplectic geometry.
result Key theorems linking classical and generalized Hamiltonian systems.
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…
The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek poly…
The aim of this paper is to define a homology theory for racks with finite rank N and use it to define invariants of knots generalizing the CJKLS 2-cocycle invariants related to the invariants defined in [15]. For this purpose, we prove that N -degenerate chains form a sub-complex of the classical complex defining rack…
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 prove that the so-called t algebra of braids and ties supports a Markov trace. Further, by using this trace in the Jones' recipe, we define invariant polynomials for classical knots and singular knots. Our invariants have three parameters. The invariant of classical knots is an extension of the Homflypt polynomial a…
In 1993, Fenn, Rourke and Sanderson introduced rack spaces and rack homotopy invariants, and modifications to quandle spaces and quandle homotopy invariants were introduced by Nosaka in 2011. In this paper, we define the Cayley-type graph and the extended quandle space of a quandle in analogy to rack and quandle spaces…
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 article surveys our ongoing project about the relationship between invariants extending the classical Rohlin invariant of homology spheres and those coming from 4-dimensional (Yang-Mills) gauge theory. The main conjecture towards which this project is directed is that the Rohlin invariant and the gauge theoretic i…
We introduce the multiplexing of a crossing, replacing a classical crossing of a virtual link diagram with multiple crossings which is a mixture of classical and virtual. For integers and an ordered -component virtual link diagram , a new virtual link diagram is ob…
Constructs maps on skein modules using non-semisimple quantum invariants.
problem Constructing maps on skein modules with specific characters.
method Uses non-semisimple invariants of 3-manifolds.
result Maps with any possible abelian non-central character as classical shadow.
New quantum invariants for 3-manifolds with boundary calculated.
problem Quantum invariants of compact 3-manifolds with non-vacuous boundary.
method Virtual link diagrams and surgery on manifolds of the form .
result Explicit calculations of invariants for some examples.
New invariant for knotted tori, similar to classical invariant.
problem Defining a new topological invariant for knotted tori.
method Analogous to Levine-Tristram invariant, using gauge theory for singular connections.
result Invariant matches Echeverria's invariant and Langte Ma's general result.
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 braid representations using virtual knot theory.
problem No classical features in virtual knot theory.
method Construct new braid representations using virtual knot theory.
result New representations of classical braids.
The paper studies invariant Einstein metrics on Lie supergroups.
problem Investigating invariant Einstein metrics on Lie supergroups.
method Parameterized left invariant metrics, derived Levi-Civita connection and Ricci tensor, reduced Einstein condition to algebraic system.
result Most real basic classical Lie superalgebras admit at least two distinct Einstein metrics.
Paper unifies three invariants for flat bundles over surfaces with boundary.
problem Measuring flat bundles over surfaces with boundary.
method Extending Atiyah-Patodi-Singer rho invariant to discontinuous class functions.
result Unified signature, Toledo, and rho invariants.
In this paper we announce the existence of a family of new -variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma-Hecke algebra of type . Yokonuma-Hecke algebras are generalizations of Iwahori-Hecke algebras, and this family contains the Homflypt polynomial, the fa…
We construct BCOV invariant for Calabi-Yau pairs. The construction covers the classical BCOV invariant and certain equivariant BCOV invariant. The BCOV invariant obtained is expected to be well-behaved under birational equivalence.
Biquandle Virtual Bracketsmath.GT
We introduce an infinite family of quantum enhancements of the biquandle counting invariant we call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored oriented knot and link diagrams with values in a commutative ring using virtual crossings as smoothings, these invariants take the…
Joyce observed that the Alexander invariant and the medial quandle of a classical knot are equivalent to each other, as invariants. In the present paper, we discuss the rather complicated extension of Joyce's observation to several different medial quandles and reduced (one-variable) Alexander modules associated with c…
Zh-construction links virtual links to classical ones, simplifying knot invariants.
problem Linking virtual and classical knot invariants.
method Zh-construction maps virtual link diagrams to classical ones.
result Zh-construction unifies and characterizes knot invariants.
We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is known that the pair of them is a complete invariant for classical knots. In this paper, we compare a situation in surface-knot theory with that in classical k…
Paper conjectures Links-Gould invariant generalizes Alexander polynomial.
problem Classifying knots and links using the Links-Gould invariant.
method Analyzing classical properties of the Links-Gould invariant.
result Evidence suggests Links-Gould invariant provides lower bounds for genus and fiberedness criteria.
Paper introduces a new invariant for planar knotoids.
problem Defining an invariant for planar knotoids.
method Using Gauss diagrams and transcendental functions.
result The invariant is a Vassiliev invariant of order one.
Researchers resolved ambiguities in gravitational radiation charges.
problem Ambiguities in charges related to gravitational radiation.
method Addressed supertranslation ambiguities in classical and extended BMS algebras.
result Proposed and proved an invariant charge free from supertranslation ambiguity.
Enhances knot invariants using bilinear forms on vector spaces.
problem Improving classical and virtual knot invariants.
method Uses bilinear forms on vector spaces indexed by pairs of elements of a finite quandle.
result New enhanced invariants of knots and links.
New polynomial invariants for virtual links are stronger than F-polynomials.
problem Defining new invariants for virtual links.
method Introducing weight functions and a recurrent construction for new invariants.
result New polynomial invariants are stronger than F-polynomials.