Positive entropy automorphisms have virtual eigenvalues outside the unit circle.
problem Understanding the dynamics of surface automorphisms with positive entropy.
method Investigating virtual homological eigenvalues of surface automorphisms.
result Existence of virtual eigenvalues outside the unit circle for automorphisms with positive entropy.
New bounds on homological eigenvalues relate to Weil-Petersson length.
problem Bounding growth of homological eigenvalues for pseudo-Anosov automorphisms.
method Established inequality linking homological Jensen square sum to Weil-Petersson translation length.
result Homological Jensen square sum grows at most linearly with covering degree compared to Weil-Petersson translation length.
Paper proves fundamental groups of certain 3+ manifold are virtually free.
problem Understanding fundamental groups of manifolds with specific curvature properties.
method Analyzing the fundamental group of closed manifolds with (n-1)-positive Ricci curvature.
result Fundamental groups of closed manifolds with 2-positive Ricci curvature are virtually free.
In this note, we show that, if a pseudo-Anosov map φ:S→S admits a finite cover whose action on the first homology has spectral radius greater than 1, then the monodromy of any fibered structure of any finite cover of the mapping torus Mφ has the same property.
We examine three key conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b_1 conjecture and the virtually fibred conjecture. We explore the interaction of these conjectures with the following seemingly unrelated areas: eigenvalues of the Laplacian, and Heegaard splittings. We first gi…
Study instantons on G2-manifolds, proving uniqueness.
problem Deforming G2-instantons on asymptotically conical manifolds. method Spinorial approach linking deformations to Dirac operators.
result Uniqueness of G2-instantons on R7. Witten deformation connects manifold spectra to Morse functions.
problem Understanding spectral properties of Riemannian manifolds.
method Rellich-Kato theorem applied to Witten deformation.
result Relates spectral package to Morse complex and harmonic oscillators.
New virtualized Δ-move simplifies virtual knots and links.
problem Simplifying virtual knots and links.
method Introducing a new local deformation called the virtualized Δ-move.
result Virtualized Δ-move is an unknotting operation for virtual knots.
Virtual index cocycles reformulate virtual link invariants.
problem No specific problem stated; focuses on reformulation.
method Using virtual index cocycles to reformulate invariants.
result Unified reformulation of virtual link invariants.
The paper studies strict equivalence in multi-virtual linkoids with new invariants.
problem Understanding strict equivalence in multi-virtual linkoids.
method Utilizing multi-virtual knot theory, defining strict virtual linkoids, and studying invariants.
result New invariants for strict virtual linkoids are defined.
New method uses Riemannian geometry to describe molecular shapes.
problem Predicting drug-like molecules using shape similarity.
method Riemannian geometry applied to molecular surfaces.
result RGMolSA method captures molecular shape effectively.
The study enumerates virtual quandles up to isomorphism.
problem Classifying virtual quandles up to isomorphism.
method Computer search and classification based on conjugacy class structures of rack automorphism groups.
result Classifications of virtual racks and quandles up to order 8.
A new method for virtual homotopy classes of virtual strings.
problem Addressing an open problem of Turaev regarding virtual homotopy classes of virtual strings.
method Generalizing Turaev's method to multistring based matrices for virtual n-strings.
result Construction of similar invariants for virtual homotopy classes of virtual n-strings.
The paper extends CF-moves to classify virtual links of any number of components.
problem Classifying virtual links using CF-moves.
method Extending CF-moves to classify virtual links of arbitrary number of components using the virtual linking number and invariants.
result Classification of 3-component even virtual links up to CF-moves.
New virtual version of Thompson's group created to handle virtual knots.
problem Creating a mathematical framework for virtual knots.
method Defining a virtual version of Thompson's group and proving its properties.
result Any virtual link can be constructed from an element of the new virtual Thompson's group.
Refines virtual link equality criterion for diagrams with one virtual crossing.
problem Determining when a virtual link diagram represents a properly virtual link.
method Refines the Kauffman-Murasugi-Thislethwaite type inequality for virtual links.
result Criterion for virtual link diagrams with exactly one virtual crossing to represent a properly virtual link.
Extend writhe polynomial from virtual knots to multi-virtual knots
problem Extend writhe polynomial from virtual knots to multi-virtual knots
method Extend writhe polynomial from virtual knots to multi-virtual knots
result Several questions asked in [16] have been answered
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.
Study virtualized Delta, Sharp, and Pass moves for oriented virtual knots and links.
problem Conditions for unknotting oriented virtual knots and links.
method Local moves (Delta, Sharp, Pass) for oriented virtual links.
result Lower bounds for the unknotting numbers are proven and shown to be best possible.
The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot obtained from the standard (p,q)-torus knot diagram by replacing all crossings o…
This paper connects virtual biquandles to biquandles for virtual link colorings.
problem Extending invariants from biquandles to virtual biquandles.
method Establishing equivalence between two representations of virtual braid groups and introducing new labeling rules.
result The number of colorings of a virtual link by virtual biquandles can be recovered from colorings by biquandles.
Method converts virtual link diagrams to normal form, preserving equivalence.
problem Normalizing virtual link diagrams to study their properties.
method Method converts virtual link diagrams to normal form.
result Normal virtual link diagrams from equivalent diagrams are equivalent.
Virtual knots and links get multicrossings and petal diagrams.
problem Defining multicrossings for virtual knots and links.
method Extending multicrossings to virtual knots and links, showing petal diagrams exist.
result Petal diagrams exist for all virtual knots.
Introduces virtual tribrackets for knot coloring.
problem Coloring regions in virtual knot diagrams.
method Algebraic structure (virtual tribrackets) for knot coloring.
result Invariant can distinguish certain virtual knots.
A virtual string can be defined as an equivalence class of planar diagrams under certain kinds of diagrammatic moves. Virtual strings are related to virtual knots in that a simple operation on a virtual knot diagram produces a diagram for a virtual string. In this paper we consider three operations on a virtual string …
Wirtinger number equals virtual bridge number for virtual links.
problem Calculating the virtual bridge number of virtual links.
method Algorithmically computing the minimum number of generators of the link group.
result The Wirtinger number equals the virtual bridge number for virtual links.
New algebraic structure for distinguishing twisted virtual handlebody-links.
problem Distinguishing twisted virtual handlebody-links.
method Introducing twisted virtual bikeigebras and using them to define invariants.
result New invariants distinguish some twisted virtual handlebody-links.
The paper defines a chord index for virtual knots and links.
problem Defining a chord index for virtual knots and links.
method Smoothing real crossing points of virtual knot diagrams to construct a chord index.
result Several polynomial invariants can be derived from this construction.
Given a virtual knot K, we construct a group VGK called the virtual knot group, and we use the elementary ideals of VGK to define invariants of K called the virtual Alexander invariants. For instance, associated to the k=0 ideal is a polynomial HK(s,t,q) in three variables which we call the virtual Alexa…
New invariant for virtual n-links defined and studied.
problem Detecting virtual trefoil and other virtual links.
method Defining a new virtual link invariant VD(K) and studying its geometric properties. result The Dehn space DD(K) is an invariant of K and can detect the virtual trefoil. 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.
Study algebraic invariants of multi-virtual links.
problem Algebraic invariants of multi-virtual links.
method Determining moves, introducing operator quandles, constructing invariants.
result Established new invariants for multi-virtual links.
Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer g…
Paper detects checkerboard colorability of virtual links using odd writhe and arrow polynomial.
problem Detecting checkerboard colorability of virtual links.
method Using odd writhe and arrow polynomial.
result Proves 6 virtual knots are not checkerboard colorable.
Virtual knot theory, introduced by Kauffman, is a generalization of classical knot theory of interest because its finite-type invariant theory is potentially a topological interpretation of Etingof and Kazhdan's theory of quantization of Lie bi-algebras. Classical knots inject into virtual knots, and flat virtual knots…
The paper discusses colorings and doubled colorings of virtual doodles.
problem Coloring virtual doodles using a new algebraic structure.
method Introduced a new algebra called a doodle switch and defined an invariant for virtual doodles.
result Introduced doubled colorings and defined an invariant for virtual doodles.
The aim of the present paper is to prove that the minimal number of virtual crossings for some families of virtual knots grows quadratically with respect to the minimal number of classical crossings. All previously known estimates for virtual crossing number were principally no more than linear in the number of classic…
A virtual link can be understood as a link in a trivial I-bundle over an orientable compact surface with genus. A twisted virtual link is a link in a trivial I-bundle over a not-necessarily orientable compact surface. A twisted virtual birack is an algebraic structure with axioms derived from the twisted virtual Reidem…
We introduce a theory of virtual Legendrian knots. A virtual Legendrian knot is a cooriented wavefront on an oriented surface up to Legendrian isotopy of its lift to the unit cotangent bundle and stabilization and destablization of the surface away from the wavefront. We show that the groups of Vassiliev invariants of …
Study virtual braid groups, proving a key subgroup result.
problem Understanding congruence subgroups in virtual braid groups.
method Using an extension of the integral Burau representation.
result Proved level 2 congruence subgroup of virtual braid group is pure virtual braid group.
Gauss codes help uniquely identify virtual doodles.
problem Identifying virtual doodles uniquely.
method Introduced left canonical Gauss codes.
result Oriented virtual doodles uniquely presented by left canonical Gauss codes.
The paper characterizes crystallographic groups derived from virtual braid and twin groups.
problem Characterizing crystallographic groups from virtual braid and twin groups.
method Analyzing quotients of virtual braid and twin groups by their commutator subgroups.
result The quotients of virtual braid and twin groups by their commutator subgroups are crystallographic groups.
The writhe polynomial invariant is proven for virtual knots via shell moves.
problem Determining equivalence of oriented virtual knots using writhe polynomials.
method Introducing shell moves to prove equivalence of writhe polynomials.
result Two virtual knots are equivalent if they can be transformed by shell moves.
Paper explores algebraic, topological, and combinatorial properties of singular virtual braids.
problem Understanding singular virtual braids and their properties.
method Algebraic relations, topological and combinatorial bijections, presentations.
result A bijection between singular abstract braids and singular virtual braids, leading to a presentation of the singular pure virtual braid monoid.
Invariants for trivalent graphs using algebraic colorings.
problem Creating an invariant for virtual trivalent spatial graphs.
method Colorings by virtual Niebrzydowski algebras.
result Generalization and computational implementation of invariants.
Paper introduces a new skein relation for multivariable polynomials of virtual links.
problem Understanding properties of virtual links through polynomial invariants.
method Developed a virtual skein relation for multivariable polynomials of virtual links.
result New skein relation for multivariable polynomials of virtual links.
Study Vassiliev invariants for virtual knots, expanding quantum theory.
problem Understanding Vassiliev invariants for virtual knots.
method Define chord diagrams, weight systems, and Lie algebra weight systems for rotational virtual knots.
result Extended quantum invariants capture more information than standard invariants.
This paper studies virtual knots using mosaic diagrams.
problem Understanding virtual knots through mosaic diagrams.
method Developed moves to preserve knot type and showed all virtual knots can be represented.
result Any virtual knot can be represented as a virtual mosaic.