Study on hyperbolic knotoids, proving their volumes add and providing tables.
problem Defining and studying hyperbolic knotoids.
method Definitions and proofs for hyperbolicity of spherical and planar knotoids, including volume calculations.
result Volumes of hyperbolic spherical knotoids add and rational knotoids have least volume.
The study extends knot theory to knotoids using two approaches.
problem Extending Vassiliev invariants to knotoids.
method Two approaches: 1) Closures to knots, 2) Directly on knotoids.
result Non-trivial type-1 invariants for spherical knotoids.
New polynomials detect non-rotatable knotoid shapes.
problem Detecting non-rotatable knotoid shapes.
method Defined homotopy index polynomials for knotoids.
result Homotopy polynomials detect non-rotatable spherical knotoids.
Paper refines braidoid equivalence for spherical knotoids.
problem Equivalence of knotoid diagrams in spherical space.
method Refined L-equivalence of braidoid diagrams. result Equivalence theorem for multi-knotoid diagrams in S2. Homological invariant for knotoids provides sharp crossing number bounds and sufficient conditions.
problem Developing an invariant for knotoids similar to Casson's invariant for knots.
method Direct and homology extension of knotoid diagrams in surfaces.
result Sharp lower bound on crossing number and sufficient condition for proper knotoids.
Complete classification of knotoids up to seven crossings.
problem Classifying spherical knotoids up to a certain number of crossings.
method Enumerating diagrams, simplifying them, distinguishing equivalence classes using various invariants.
result Conjecture that classification up to seven crossings is complete.
Modified knotoids with framing and coframing for quantum invariants.
problem Defining and classifying knotoids with framing.
method Defining framed and biframed knotoids, showing topological correspondence, and constructing quantum invariants.
result Generalized quantum knotoid invariants constructed.
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.
The Thistlethwaite theorem is extended to knotoids and linkoids.
problem Extending a classical theorem to a new class of knots and links.
method Defining new invariants and associated polynomials for knotoids and linkoids.
result The Thistlethwaite theorem is proven for knotoids and linkoids.
The paper explores knotoids, pseudo knotoids, braidoids, and pseudo braidoids on the torus.
problem The study of knotoids, pseudo knotoids, braidoids, and pseudo braidoids on the torus.
method Introducing new knotoid and braidoid concepts, isotopy theorems, state sum formulas, and Alexander and Markov theorems.
result Formulation and proof of Alexander and Markov theorems for mixed knotoids and mixed pseudo knotoids.
The paper extends knotoid theory to annular and toroidal settings.
problem Extending knotoid theory to new geometric settings.
method Introducing new knotoid types, extending bracket polynomials.
result Universal bracket polynomials for annular and toroidal knotoids.
Geometrically interprets virtual knotoids in thickened surfaces.
problem No specific problem stated; focuses on geometric interpretation.
method Geometric interpretation of virtual knotoids as arcs in thickened surfaces.
result Shows virtual knotoid theory as a generalization of classical knotoid theory.
Invariants derived for rail knotoids based on associated knots.
problem Deriving invariants for rail knotoids.
method Associated two unoriented and oriented knots, then translated to rail isotopy invariants.
result Derived invariants for rail knotoids.
We introduce and study knotoids. Knotoids are represented by diagrams in a surface which differ from the usual knot diagrams in that the underlying curve is a segment rather than a circle. Knotoid diagrams are considered up to Reidemeister moves applied away from the endpoints of the underlying segment. We show that kn…
New quantum invariants for planar knotoids improve knot classification.
problem Classifying and distinguishing planar knotoids with up to five crossings.
method Define biframed planar knotoids and construct new invariants.
result Improved classification of planar knotoids with up to five crossings.
The paper studies unknotting operations and numbers for plus-welded knotoids.
problem Understanding unknotting operations and numbers for plus-welded knotoids.
method The paper proves transformations and introduces new operations to calculate unknotting numbers.
result Upper bounds for unknotting numbers of plus-welded knotoids are found.
Knotoids were introduced by V. Turaev as open-ended knot-type diagrams that generalize knots. Turaev defined a two-variable polynomial invariant of knotoids which encompasses a generalization of the Jones knot polynomial to knotoids. We define a triply-graded homological invariant of knotoids categorifying the Turaev p…
The paper constructs quantum invariants for knotoid diagrams.
problem Quantum invariants for knotoid diagrams in R2. method Decompose Morse knotoid diagrams into basic elementary diagrams, each associated with a matrix solving the quantum Yang-Baxter equation. Define quantum state sum models to recover various polynomials.
result Recover and define new polynomials for Morse knotoids.
Clock theorem extended to knotoids and linkoids.
problem Generalizing the Clock Theorem to knotoids and linkoids.
method Extending the Clock Theorem to knotoids and linkoids.
result Clock states of knotoid diagrams form a lattice under transpositions.
Researchers extend Alexander polynomial to knotoids and linkoids.
problem Defining and studying Alexander polynomial extensions for knotoids and linkoids.
method Developed and proved conjecture on mock Alexander polynomial for knotoids and linkoids.
result Proved conjecture on mock Alexander polynomial for knotoids and linkoids.
In this paper, we consider biquandle colorings for knotoids in R2 or S2 and we construct several coloring invariants for knotoids derived as enhancements of the biquandle counting invariant. We first enhance the biquandle counting invariant by using a matrix constructed by utilizing the orientation a kno…
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…
New F-polynomial distinguishes knotoid diagrams not previously possible.
problem Polynomial invariants for knotoids.
method Introduced F-polynomial and constructed distinguishing examples. result New invariant F distinguishes knotoids not previously possible. Paper studies knotoid chirality using shadow quandle colorings and invariants.
problem Distinguishing knotoids from their mirrors.
method Shadow quandle colorings and cocycle invariants.
result Knotoid 31 is shown to be chiral. In this paper we generate and systematically classify all prime planar knotoids with up to 5 crossings. We also extend the existing list of knotoids in S2 and add all knotoids with 6 crossings.
By using double branched covers, we prove that there is a 1-1 correspondence between the set of knotoids in the 2-sphere, up to orientation reversion and rotation, and knots with a strong inversion, up to conjugacy. This correspondence allows us to study knotoids through tools and invariants coming from knot theory. In…
Signed heights of knotoids are defined and studied.
problem Understanding the signed height of knotoids.
method Defined positive and negative parts of height, proved they determine unsigned height, provided lower bounds with polynomials, studied associated sequences.
result Positive and negative parts of height determine unsigned height.
Extends knotoid theory to include multiple poles and intervals.
problem No new problem introduced.
method Definition of generalized knotoids and graphs, exploration of invariants.
result Theory subsumes various topological objects and introduces new cases.
The paper explores knotoid invariants on the plane using quandles and cocycles.
problem Classifying planar knotoids due to their complexity and lack of invariants.
method Investigates equivalence of planar knotoids using quandle colorings and cocycle invariants.
result Introduces a new invariant called the triangular quandle cocycle invariant.
Introduced coloring-allowed invariants of planar knotoids with the coloring number.
problem Construction of polynomial invariants of knotoids with signs of crossings.
method Defined coloring-allowed invariants of planar knotoids with the coloring number.
result Discussed the 4-phases functions of coloring-allowed invariants.
The paper establishes a relation between knotoid crossing number and height.
problem Establishing a relation between knotoid crossing number and height.
method Using Reidemeister moves and isotopies, the paper proves a relation between the crossing number and height of a knotoid.
result The crossing number of a knotoid is at least twice its height.
Introduce a two-variable parity polynomial for virtual knotoids
problem Define a polynomial invariant for virtual knotoids
method Based on the parity of classical crossings
result Can distinguish pairs not distinguished by odd writhe and affine index polynomial
This paper is a survey on the theory of knotoids and braidoids. Knotoids are open ended knot diagrams in surfaces and braidoids are geometric objects analogous to classical braids, forming a counterpart theory to the theory of knotoids in the plane. We survey through the fundamental notions and existing works on these …
Study of pseudo knots, links, and knotoids with braiding and L-moves.
problem Missing crossing information in knots and links.
method Introducing pseudo knots, pseudo tied links, pseudo knotoids, and L-moves.
result Formulated new theorems for pseudo knots, links, and knotoids.
New framework distinguishes knots via neighborhood invariants.
problem Distinguishing knots and knotoids.
method Study of knotoid spectra and neighborhood invariants.
result Neighborhood invariants can distinguish knots of higher Gordian distance.
Paper introduces a new invariant for virtual knotoids and proves it's a Vassiliev invariant of order one.
problem Tackles the problem of understanding invariants for virtual knotoids.
method Uses a 0-smoothing invariant constructed from local modifications at classical crossings.
result Demonstrates that the 0-smoothing invariant provides less information than the gluing invariant.
Extends knotoid theory to multi-linkoids, studying various invariants.
problem Defining and analyzing invariants for multi-linkoids in a closed surface.
method Examines Kauffman bracket, ordered bracket, skein module, and T-invariant.
result Developed new invariants for multi-linkoids.
This paper investigates the parity concept in knotoids in S2 and in R2 in relation with virtual knots. We show that the virtual closure map is not surjective and give specific examples of virtual knots that are not in the image. We introduce a planar version of the parity bracket polynomial for knotoids …
We study here global and local entanglements of open protein chains by implementing the concept of knotoids. Knotoids have been introduced in 2012 by Vladimir Turaev as a generalization of knots in 3-dimensional space. More precisely, knotoids are diagrams representing projections of open curves in 3D space, in contras…
Recent studies classify the topology of proteins by analysing the distribution of their projections using knotoids. The approximation of this distribution depends on the number of projection directions that are sampled. Here we investigate the relation between knotoids differing only by small perturbations of the direc…
The paper defines and analyzes quandles of knotoids and linkoids.
problem Tackling the invariant properties of knotoids and linkoids.
method Defining fundamental quandles and pointed quandles, proving invariance, and introducing new invariants.
result Fundamental pointed quandles enhance the fundamental quandle and distinguish 1-linkoids.
We work on the notions of rail arcs and rail isotopy in R3, and we introduce the notions of rail knotoid diagrams and their equivalence. Our main result is that two rail arcs in R3 are rail isotopic if and only if their knotoid diagram projections onto the plane of the two lines which we call ra…
New findings on prime theta-curves with simple tangles.
problem Understanding prime theta-curves with specific unknotting numbers.
method Analyzing composite theta-curves and their components.
result Composite theta-curves with unknotting number one are prime.
Braidoids generalize the classical braids and form a counterpart theory to the theory of planar knotoids, just as the theory of braids does for the theory of knots. In this paper, we introduce basic notions of braidoids, a closure operation for braidoids, we prove an analogue of the Alexander theorem, that is, an algor…
The presence of slipknots in configurations of proteins and DNA has been shown to affect their functionality, or alter it entirely. Historically, polymers are modeled as polygonal chains in space. As an alternative to space curves, we provide a framework for working with subknots inside of knot diagrams via knotoid dia…
The backbone of most proteins forms an open curve. To study their entanglement, a common strategy consists in searching for the presence of knots in their backbones using topological invariants. However, this approach requires to close the curve into a loop, which alters the geometry of curve. Knoto-ID allows evaluatin…
Study of bonded knots and braids with new algebraic models.
problem Classifying and understanding physical or chemical bonds in knots and braids.
method Developed new algebraic models (bonded knots, braids, braidoids) and invariants.
result New algebraic structures and invariants for bonded knots and braids.
New invariants measure entanglement of open curves in 3D space.
problem Characterizing the complexity of open curves in 3-space.
method Introducing linkoids and virtual knots, and new invariants.
result Strong invariants of linkoids independent of virtual closure.