Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Re…
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48 results for “Triple Crossing Diagram Maps”
Integrable dynamics explained via geometric maps and cluster algebras.
problem Integrable dynamics in projective geometry.
method Twisted triple crossing diagram maps and cluster integrable systems.
result Cross-ratio dynamics described by geometric -matrices.
Enumerates knots up to five crossings and describes moves between them.
problem Counting and classifying knots up to a specific number of crossings.
method Generated tables of minimal diagrams and derived moves between knots.
result Conjecture about a lower bound for the triple-crossing number based on Alexander polynomial.
In this paper we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known knots and links with triple-crossing number equal to five. By intro…
Formula for Milnor triple linking number in link diagrams with multiple crossings.
problem Calculating Milnor triple linking number for complex link diagrams.
method Polyak-Viro type formula with explicit computation of configuration space integral.
result Formula applicable to diagrams with triple or more crossings.
This paper shows how to create surface-links with many triple points.
problem Creating surface-links with a large number of triple points.
method Analogous to knot diagrams, the paper uses broken sheet diagrams to project surface-links and analyze their triple points.
result There are non-split surface-links with arbitrarily many triple points.
The paper improves bounds on knot crossings and tabulates minimal diagrams.
problem Improving bounds on knot crossings and tabulating minimal diagrams.
method Analyzing triple-crossing and delta-crossing numbers, proving tangle existence, generating tables.
result Improved bounds on knot crossings and tabulated minimal diagrams for prime knots up to delta-crossing number 4.
Develops TCD maps to relate discrete differential geometry and cluster algebras.
problem Capturing constraints and dynamics in discrete differential geometry.
method Triple crossing diagram maps (TCD maps) and geometric operations.
result Establishes a hierarchy of cluster structures on TCD maps.
This paper compiles and calculates triple point numbers for surface-links in Yoshikawa's table.
problem Determining the triple point number of surface-links in Yoshikawa's table.
method Using broken sheet diagrams, the paper compiles known triple point numbers and calculates or bounds the remaining ones.
result Compilation and calculation of triple point numbers for surface-links in Yoshikawa's table.
Paper introduces simplified formulas for Milnor's triple linking number.
problem Computational difficulty in calculating Jones polynomial for topological polymers.
method Developed Gauss diagram formulas for Milnor's Vassiliev invariants.
result Introduced non-torsion valued Milnor's triple linking number.
Unified framework for various geometric constructions.
problem Organizing diverse geometric constructions.
method Introducing TCD maps and defining local moves.
result Two distinct cluster structures on TCD maps.
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.
New method constructs moduli spaces of Lagrangian surfaces in CP^2 from grid diagrams.
problem Constructing explicit examples of triple grid diagrams for Lagrangian surfaces in CP^2.
method Elegant geometric construction reducing to linear algebra.
result Explicit construction of moduli space of triple grid diagrams.
Formula for computing triple-cup product from Heegaard diagrams of 3-manifolds.
problem Computing the triple-cup product invariant of 3-manifolds.
method Explicit formula from Heegaard diagrams and reduction of Turaev's homotopy intersection form.
result Triple-cup product can be recovered from Heegaard diagrams and Turaev's form.
Paper shows how to represent Milnor's triple linking number using chord diagrams and doodle invariants.
problem Tackles the representation of Milnor's triple linking number.
method Establishes an analogous description for Milnor's triple linking number using counts of chord diagrams and doodle invariants.
result Shows that Milnor's triple linking number can be represented in terms of chord diagrams and doodle invariants.
Roseman moves are seven types of local modification for surface-link diagrams in -space which generate ambient isotopies of surface-links in -space. In this paper, we focus on Roseman moves involving triple points, one of which is the famous tetrahedral move, and discuss their independence. For each diagram of an…
Paper explores relationships between triple chords and a specific homotopy relation in knot theory.
problem Understanding the relationship between triple chords and a homotopy equivalence class in knot theory.
method Analyzes the number of triple chords and their connection to the strong (1, 2) homotopy equivalence class.
result Prime knot projections are trivialized by strong (1, 2) homotopy if they have no triple chords.
A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c_3(…
We consider diagrams of links in obtained by projection from with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots are exhibited. In particular, we establish which of these knots are algebraic an…
The singularity set of a generic standard projection to the three space of a closed surface linked in four space, consists of at most three types: double points, triple points or branch points. We say that this generic projection image is p-diagram if it does not contain any triple point. Two p-diagrams of equivalent s…
Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A triple crossing is a crossing where three strands meet at a single point, such that each strand bisects the crossing. In this paper we find a relationship between the triple crossing number and th…
A simplified trisection is a trisection map on a 4-manifold such that, in its critical value set, there is no double point and cusps only appear in triples on innermost fold circles. We give a necessary and sufficient condition for a 3-tuple of systems of simple closed curves in a surface to be a diagram of a simplifie…
Triple-crossing number bound for knots and links, especially torus knots.
problem Finding bounds for triple-crossing numbers of knots and links.
method Using the genus of a knot or link, we derive bounds for the triple-crossing number.
result Triple-crossing number of torus knots and many other knots is at least twice their genus.
Harmonic unit normal sections studied for Grassmannians induced by cross products.
problem Energy of maps assigning unit vectors to subspaces of Grassmannians.
method Analyzing cross products to induce harmonic sections into sphere bundles.
result All unit normal sections of Grassmannians associated with cross products are harmonic.
The paper shows that knot projections without triple chords can be simplified.
problem The study of knot projections and their chord diagrams.
method Flat Reidemeister moves that decrease 1-gons or strong 2-gons.
result For any knot projection without triple chords, a sequence of moves simplifies it to a simple closed curve.
Researchers create spectral triples for twisted crossed products using Kasparov's external product.
problem Constructing spectral triples for twisted crossed products.
method Using Kasparov's external product, the construction of spectral triples for twisted crossed products is achieved.
result The construction of spectral triples for twisted crossed products is possible under suitable assumptions.
We describe a model of random links based on random 4-valent maps, which can be sampled due to the work of Schaeffer. We will look at the relationship between the combinatorial information in the diagram and the hyperbolic volume. Specifically, we show that for random alternating diagrams, the expected hyperbolic volum…
A formula that relates triple points, branch points, and their distances from infinity is presented. We recover trivial normal Euler classes for oriented surfaces, and formulas on signed triple points.
Let be a finite group. Noncommutative geometry of unital -algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical o…
We construct various functorial maps (projections) from virtual knots to classical knots. These maps are defined on diagrams of virtual knots; in terms of Gauss diagram each of them can be represented as a deletion of some chords. The construction relies upon the notion of parity. As corollaries, we prove that the mini…
New invariant calculates 4-manifolds using trisection diagrams and combings.
problem Calculating non-semisimple 4-manifold invariants.
method Using trisection diagrams and combings of the trisection surface.
result Invariant calculated for Stein nuclei, generalizing earlier semisimple version.
Introduces a new triple coproduct for knots on surfaces, preserving local crossing patterns.
problem Tackles the lack of fine-grained detection in classical cobrackets for local crossing patterns.
method Defines an integer-valued invariant using a coproduct and intersection theory, extending Turaev's cobracket theory.
result Reveals an intrinsic simplicity in the algebraic framework, uniquely determining relations in the word space.
We describe some regular techniques of calculating finite degree invariants of triple points free smooth plane curves . They are a direct analog of similar techniques for knot invariants and are based on the calculus of {\em triangular diagrams} and {\em connected hypergraphs} in the same way as the calcul…
Extends Manin triples to Lie bialgebroids over Lie groupoids.
problem Characterizing Lie bialgebroids via Manin triples.
method Establishing correspondence between Lie bialgebroid groupoids and multiplicative Manin triples.
result New viewpoint on co-quadratic Lie algebroids and Manin triple description of Lie bialgebroid crossed modules.
Proves minimal crossing diagrams for specific spatial graphs.
problem Proving minimal crossing diagrams for spatial graphs.
method Analyzing adequate diagrams and replacing vertices and edges.
result All 1-vertex spatial graphs with adequate diagrams have minimal crossing number.
The paper explores when specific knot operations simplify diagrams.
problem Understanding when arc crossing changes simplify knot diagrams.
method Examined two types of arc crossing changes on link diagrams and determined when they are unknotting operations.
result Any two crossing points in an alternating knot diagram are arc crossing change admissible.
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.
A plane curve is a knot diagram in which each crossing is replaced by a 4-valent vertex, and so are dual to a subset of planar quadrangulations. The aim of this paper is to introduce a new tool for sampling diagrams via sampling of plane curves. At present the most efficient method for sampling diagrams is rejection sa…
Minimal grid diagrams for 12-crossing prime knots identified.
problem Identifying minimal grid diagrams for prime knots.
method Listed minimal grid diagrams for 12-crossing prime knots.
result Provided a list of minimal grid diagrams for 12-crossing prime knots.
The paper develops a method to map knots in a cylinder to virtual-flat knots.
problem How to map knots in a cylinder to virtual knots.
method Construct a diagram on a cylinder with invisible crossings, then pull back invariants.
result Developed a method to map knots in a cylinder to virtual-flat knots.
New estimate of semimeander complexity for knots with more than 10 crossings.
problem Estimating the complexity of semimeander diagrams of knots.
method Proved a new upper bound on the number of crossings for semimeander diagrams of knots with more than 10 crossings.
result For knots with more than 10 crossings, semimeander diagrams have no more than crossings.
The paper describes new types of picture-valued invariants and their applications.
problem Understanding and categorizing picture-valued invariants on diagrams.
method General description and classification of derivations and functorial maps.
result Two new examples of functorial maps are introduced, including the order and lifting maps.
The study improves inequalities for link diagrams and introduces weak rectangular diagrams.
problem Improving inequalities for link diagrams and understanding their properties.
method Introducing weak rectangular diagrams and proving new inequalities.
result Generalizes and subsumes many known inequalities related to multi-crossing numbers.
Paper classifies link diagrams on nonorientable surfaces using region crossing changes.
problem Classifying link diagrams on nonorientable surfaces.
method Classification through region crossing changes.
result Classification of link diagrams on nonorientable surfaces.
A cobordism realizing crossing change on tangle homology and a categorified Vassiliev skein relationmath.GT
The paper introduces a cobordism for Khovanov homology crossing change and categorifies Vassiliev skein relations.
problem Khovanov homology and crossing changes in tangle diagrams.
method Introducing a sum of cobordisms that yields a morphism on Khovanov homology complexes for crossing change.
result The introduced cobordism is invariant under double point moves and categorifies Vassiliev skein relations.
New invariant from quantum algebra for 3-manifold bundles.
problem Quantum invariants of flat 2-bundles over 3-manifolds.
method From an involutory Hopf algebra graded by a crossed module, constructing a homotopy invariant via -colored Heegaard diagrams.
result Reduces to Kuperberg invariant when bundle is trivializable.
We consider a natural model of random knotting- choose a knot diagram at random from the finite set of diagrams with n crossings. We tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type, allowing us to compute exact probabilities for knots in this model. As expected, most diagrams with 1…
Paper introduces new link homotopy invariants and applies them to 3-bouquet graphs.
problem Milnor's triple linking number and its applications in link homotopy.
method Developed new integer-valued link homotopy invariants and applied them to 3-bouquet graphs.
result Found new integer-valued invariants derived from four terms summing to Milnor's triple linking number.