New method answers open question on symmetric diagrams.
problem Symmetric equivalence of symmetric union diagrams.
method Refined topological spin models.
result Complete answer to open question.
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48 results for “symmetric union diagrams”
An open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of such representations. Eisermann and Lamm investigated the latter question by introducing a notion of symmetric equivalence…
Table of symmetric diagrams for knots up to 10 crossings.
problem Finding symmetric diagrams for strongly invertible knots.
method Compilation of symmetric diagrams for knots up to 10 crossings.
result Similarity of transversal diagrams to symmetric union diagrams for strongly invertible knots.
Study shows not all ribbon knots can be symmetric unions.
problem Whether every ribbon knot can be a symmetric union.
method Exhibited a specific ribbon Montesinos knot that cannot be a symmetric union.
result Found a ribbon knot that is not a symmetric union.
Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka 50 years ago. It is easy to see that every symmetric union represents a ribbon knot, but the converse is still an open problem. Besides existence it is natural to consider the question of un…
We present the results of Axel Seeliger's tabulation of symmetric union presentations for ribbon knots with crossing numbers 11 and 12 and exhibit possible examples for ribbon knots which are not representable as symmetric unions. In addition, we give a complete atlas of band diagrams for prime ribbon knots with 11 and…
Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. For symmetric diagrams we develop a two-variable refinement of the Jones polynomial that is invariant under symmetric Reidemeister moves. Here the two variables and $…
Symmetric ribbon disksmath.GT
We study the ribbon discs that arise from a symmetric union presentation of a ribbon knot. A natural notion of symmetric ribbon number is introduced and compared with the classical ribbon number. We show that the gap between these numbers can be arbitrarily large by constructing an infinite family of ribbon knots with …
New knots found with same determinant but no symmetric relation.
problem Determining if knots with the same determinant are symmetrically related.
method Constructing a family of knots with the same determinant but no symmetric relation.
result No two knots in the family are symmetrically related.
The paper defines a preorder on links and explores its implications for symmetric unions.
problem Understanding the relationships between links and their symmetric unions.
method Defining a preorder relation and proving properties of links and their orbifold groups.
result If a link is a Montesinos link with rational tangles, then is either a Montesinos link with at most rational tangles or a certain connected sum.
The twisting number of a ribbon knot is at least as large as its doubly slice genus.
problem Proving a lower bound for the twisting number of ribbon knots in terms of their doubly slice genus.
method Analyzing symmetric unions and tangle replacements to establish the bound.
result Ribbon knots have arbitrarily high twisting numbers, matching their doubly slice genus.
Study on symmetric braid index of ribbon knots, deriving bounds and characterizations.
problem Understanding the symmetric braid index of ribbon knots.
method Defining symmetric braid index, using Khovanov homology, and calculating bounds.
result Existence of knots with symmetric braid index greater than braid index.
Counting geodesics on compact symmetric spaces using orbit dimensions and topological data.
problem Counting geodesics on compact symmetric spaces.
method Using orbit dimensions and topological data of the symmetric space.
result Obtained data on dimensions and connected components of focal orbits.
Study of symmetric unions of knots with new inequality and epimorphism results.
problem Understanding the genera of symmetric unions of knots.
method Introduced symmetric unions inspired by earlier work, showed an identity between twisted Alexander polynomials and genera, and established an epimorphism between knot groups.
result Obtained an inequality concerning the genera of symmetric unions and provided a positive answer to an old problem.
Tanaka shows amphichiral symmetric unions of the unknot are trivial.
problem Understanding amphichiral symmetric unions and their Jones polynomials.
method Analyzing the Jones polynomial of amphichiral symmetric unions of the unknot and generalizing to other knots.
result Amphichiral symmetric unions of any knot with one twist region are trivial.
We prove that all 2-bridge ribbon knots are symmetric unions.
Extended symmetric unions extend properties of Alexander polynomials.
problem Properties of Alexander polynomials of symmetric unions.
method Constructing pairs of knots with specific epimorphisms.
result Alexander polynomials of constructed knots exhibit extended properties.
A short proof for a theorem about composite knots.
problem Proving a theorem about composite knots with symmetric union presentations.
method Presenting a concise proof of Tanaka's theorem.
result Composite knots with symmetric union presentations have non-trivial connected summands.
On two-bridge ribbon knotsmath.GT
Two-bridge ribbon knots have symmetric union presentations.
problem Characterizing two-bridge ribbon knots.
method Symmetric union presentations and partial knot analysis.
result Symmetric union presentations for various two-bridge ribbon knots.
The article enumerates doubly symmetric diagrams for knots up to 18 crossings.
problem Enumerating doubly symmetric diagrams for knots.
method Developed an enumeration strategy for prime knots given by doubly symmetric diagrams.
result Determined all cases of doubly symmetric diagrams up to 18 crossings.
Extended symmetric union with multiple tangle regions and Alexander polynomial properties.
problem Characterizing knots with multiple tangle regions.
method Generalizing the symmetric union construction to include multiple tangle regions and analyzing the Alexander polynomial.
result The Alexander polynomial of the constructed knot is the product of the Alexander polynomials of the tangles and the square of the partial knot's Alexander polynomial.
Paper confirms Kashaev's signature conjecture for links.
problem Proving Kashaev's conjecture about link invariants.
method Using Seifert surface definition and diagrammatic approach.
result Established Kashaev's conjecture, providing a new formula for Alexander polynomial.
Paper classifies compact symmetric triads using double Satake diagrams and canonical forms.
problem Classifying compact symmetric triads.
method Introducing double Satake diagrams and canonical forms, proving their existence and properties.
result Existence and properties of canonical forms for compact simple symmetric triads.
New divide with gleams method simplifies symmetric link representation.
problem Representing symmetric links with divides and gleams.
method Introducing divides with gleams and algorithms for transvergent diagrams.
result Divides with gleams simplify the representation of symmetric links.
This paper investigates symmetric ribbon numbers of low-complexity knots.
problem Determining the minimum number of ribbon singularities in symmetric ribbon disks for knots with up to 12 crossings.
method Systematic investigation using knot polynomials and determinants.
result Novel lower bounds for symmetric ribbon numbers of knots with up to 12 crossings.
Researchers describe a specific type of submanifolds in Euclidean space.
problem Understanding inhomogeneous almost symmetric submanifolds.
method Completely describing submanifolds as unions of parallel symmetric submanifolds.
result Described inhomogeneous properly embedded almost symmetric submanifolds as unions of symmetric submanifolds.
Describes automorphism group of Rauzy diagrams.
problem Understanding the structure of Rauzy diagrams.
method Using an example from Yoccoz's unpublished notes.
result Automorphism group described as a subgroup of symmetric group.
New symmetric quandles constructed from group elements and subgroups.
problem Understanding the structure of symmetric quandles.
method Constructing symmetric quandles from specific group elements and subgroups.
result Every symmetric quandle is isomorphic to the disjoint union of constructed quandles.
A symmetric union of two knots is a classical construction in knot theory which generalizes connected sum, introduced by Kinoshita and Terasaka in the 1950s. We study this construction for the purpose of finding an infinite family of hyperbolic non-fibered three-bridge knots of constant determinant which satisfy the we…
The paper defines new representations and groups related to virtual links.
problem Defining and studying new representations of virtual braid groups and link groups.
method Introducing virtually symmetric representations, virtual link groups, and marked Gauss diagrams.
result Established equivalence of many known representations to virtually symmetric representations.
By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle cocycle invariants are also defined for unoriented links and surface-links via broken …
Half grid diagrams prove every link can be represented by a special type of grid diagram.
problem Representing links using grid diagrams and related invariants.
method Defining half grid diagrams and constructing canonical pairs, proving equivalence to Jones' construction, relating to classical link invariants.
result Established a new method to relate the oriented Thompson index to classical link invariants and provided bounds for knot invariants.
Pólya's theorem extended to meromorphic functions on Riemann surfaces.
problem Distribution of zeros of iterated derivatives of meromorphic functions.
method Recasting local arguments into translation surfaces and using flat metrics.
result Asymptotic distribution of zeros on compact Riemann surfaces.
Study identifies prime strongly positive amphicheiral knots with double symmetry.
problem Characterizing prime strongly positive amphicheiral knots with specific symmetries.
method Examined knots up to 16 crossings, identified prime knots with double symmetry, and presented almost doubly symmetric diagrams.
result Found the first prime strongly positive amphicheiral knot not slice.
The paper introduces triple grid diagrams to construct Lagrangian surfaces in complex projective space.
problem Constructing Lagrangian surfaces in complex projective space.
method Defining and analyzing triple grid diagrams to determine Lagrangian caps and surfaces.
result Triple grid diagrams can determine closed Lagrangian surfaces in under certain conditions.
Studying the isotropy orbits of compact symmetric spaces Reiswich introduced a family of explicit polynomials in one variable in order to describe the unique minimal isotropy orbit of compact symmetric spaces with Dynkin diagram of type . Based on this geometric interpretation he conjectured that these polynomials…
Let be an oriented link diagram with the set of regions . We define a symmetric map (or matrix) that gives rise to an invariant of oriented links, based on a slightly modified -equivalence of Trotter…
A symmetric quandle is a quandle with a good involution. For a knot in \$R^3\$, a knotted surface in \$R^4\$ or an \$n\$-manifold knot in \$R^{n+2}\$, the knot symmetric quandle is defined. We introduce the notion of a symmetric quandle presentation, and show how to get a presentation of a knot symmetric quandle from a…
Loops in surfaces, chord diagrams, interlace graphs: operad factorisations and generating grammarsmath.GT
Loops in surfaces and chord diagrams are studied with graph factorizations and grammars.
problem Understanding loops in surfaces and their properties.
method Factorization of filoops into spheric and toric sums, and grammars generating chordiagraphs.
result Minimal genus of filoops and stability properties under factorizations.
Minimal grid diagrams found for 13-crossing prime knots.
problem Finding the simplest grid diagrams for prime knots with 13 crossings.
method Converted prime alternating knots to grid diagrams, focusing on minimal configurations.
result 4878 prime alternating knots with 13 crossings have been represented by grid diagrams with 15 vertical segments.
Two families of symmetric structures found in 7D.
problem Identifying symmetric structures in 7D.
method Analyzing homogeneous spaces and root diagrams of .
result Found two families of symmetric structures with distinct properties.
Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagrammath.DG
The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian symmetric space from the Satake diagram, in a way that is suited for the use with comput…
Geometrically connects Laplace eigenfunctions to Borel-Weil theory on symmetric spaces.
problem Understanding the spectral properties of Laplace-Beltrami operators on Riemannian symmetric spaces.
method Using symplectic geometry and geometric quantization, associating flag manifolds to symmetric spaces and relating their Satake diagrams.
result Harmonic polynomials on flag manifolds induce all eigenfunctions on symmetric spaces.
Basket links are shown to be isotopic to .
problem Understanding isotopy of basket links to torus links.
method Using symmetrized Seifert form congruence to matrix.
result Basket links are isotopic to .
Study of sectorial decompositions in symmetric products of surfaces for symplectic geometry.
problem Symplectic topology of symmetric products of Riemann surfaces.
method Liouville sectorial techniques.
result New geometric proof of Homological Mirror Symmetry for a specific case.
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.
Local orbit types of the isotropy representations for semisimple pseudo-Riemannian symmetric spacesmath.DG
We list up all the possible local orbit types of hyperbolic or elliptic orbits for the isotropy representations of semisimple pseudo-Riemannian symmetric spaces. It is key to give a recipe to determine the local orbit types of hyperbolic principal orbits by using three kind of restricted root systems and Satake diagram…
Given a singular Schubert variety Z in a compact Hermitian symmetric space it is a longstanding question to determine when Z is homologous to a smooth variety Y. We identify those Schubert varieties for which there exist first-order obstructions to the existence of Y. This extends (independent) work of M. Walters, R. B…