A diagrammatic language for 3D manifolds with boundary.
problem Representing and manipulating 3D manifolds with boundary.
method Diagrammatic calculus and local moves.
result Completeness of the diagrammatic calculus proved.
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48 results for “diagrammatic moves”
In this survey paper we present the --moves between braids and how they can adapt and serve for establishing and proving braid equivalence theorems for various diagrammatic settings, such as for classical knots, for knots in knot complements, in c.c.o. 3--manifolds and in handlebodies, as well as for virtual knots, …
The paper simplifies knot and link diagrams with triple-crossings.
problem Generating and classifying minimal triple-crossing knot and link diagrams.
method Systematic method to generate minimal triple-crossing projections, introducing new diagrammatic moves.
result Classification of knots and links with triple-crossing number up to five, derivation of minimal generating set of moves.
New moves for singular knots identified and described.
problem Identifying and describing moves for singular knots.
method Provided 96 generating sets of oriented singular Reidemeister moves and selected moves for Legendrian singular knots.
result Surviving moves for Legendrian singular knots were identified and described.
New contact Kirby moves complete the set for contact surgery diagrams.
problem Contact surgery diagrams and their relation to contactomorphic contact manifolds.
method Introducing lantern moves and chain moves to complete the set of contact Kirby moves.
result Two contact surgery diagrams represent contactomorphic contact manifolds if and only if they are related by a sequence of specific moves.
Using the diagrammatic calculus for Soergel bimodules developed by B. Elias and M. Khovanov, we show that Rouquier complexes are functorial over braid cobordisms. We explicitly describe the chain maps which correspond to movie move generators.
A diagrammatic presentation and its characterization of non-split compact surfaces in the 3-spheremath.GT
We give a presentation for a non-split compact surface embedded in the 3-sphere by using diagrams of spatial trivalent graphs equipped with signs and we define Reidemeister moves for such signed diagrams. We show that two diagrams of embedded surfaces are related by Reidemeister moves if and only if the surfaces repres…
In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some -moves that only differ in how the three strands that are involved in the move are ordered on the knot. To transform knot diagrams of isotopic knots into each other …
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…
Paper simplifies proof of slide-equivalence in crown diagrams.
problem Proving slide-equivalence of crown diagrams.
method Diagrammatic approach to generalized shift moves.
result Slide-equivalence of vanishing cycles in crown diagrams.
New diagrams classify triply periodic entanglements.
problem Classifying triply periodic entanglements.
method Introducing diagrams for links in 3-torus and defining additional moves.
result New diagrams allow classification of non-isotopic links with same preimage.
Crane and Frenkel proposed a state sum invariant for triangulated 4-manifolds.They defined and used new algebraic structures called Hopf categories for their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double of a finite group. I…
Study Alexander polynomials of links in 3-torus.
problem Investigate Alexander polynomials of links in 3-torus.
method Diagrammatic approach, Reidemeister moves, fundamental group, homology group, Alexander polynomials, twisted Alexander polynomials.
result Computed Alexander and twisted Alexander polynomials of links in 3-torus.
The paper computes the Kauffman bracket skein module of via braids.
problem Computing the Kauffman bracket skein module of .
method Two methods: extending a universal invariant to via braid band moves and a diagrammatic approach.
result The Kauffman bracket skein module of is not torsion-free and its free part is generated by the unknot.
We consider ribbon n-knots for n\geq 2. For such knots we define a set of moves on ribbon disks, and show that any two ribbon disks for isotopic knots are related by a finite sequence of such moves and ambient isotopies. Using this we are able to prove that there is a natural geometric correspondence between ribbon n-k…
The paper explores extensions of local moves on string links and their relation to ribbon surfaces.
problem Finding a unique extension of classical local moves on string links.
method Relating local moves to surgeries on ribbon surfaces in R^4.
result There can be at most one unique welded extension of a classical move that is a ribbon residue.
Uniform convergence of isotopies implies ambient isotopy, aiding knot equivalence.
problem Determining when uniform convergence of isotopies leads to ambient isotopies.
method Using a diagrammatic condition to offload uniform convergence, constructing examples of tame knots.
result Constructing tame knots with countably-many crossings, distinguishing them from wild curves.
When the signed weighted resolution set was defined as an invariant of pseudoknots, it was unknown whether this invariant was complete. Using the Gauss-diagrammatic invariants of pseudoknots introduced by Dorais et al, we show that the signed were-set cannot distinguish all non-equivalent pseudoknots. This goal is achi…
We introduce the notion of directed diagrammatic reducibility which is a relative version of diagrammatic reducibility. Directed diagrammatic reducibility has strong group theoretic and topological consequences. A multi-relator version of the Freiheitssatz in the presence of directed diagrammatic reducibility is given.…
Study surfaces in 4-manifolds using banded unlink diagrams.
problem Representing and manipulating immersed surfaces in 4-manifolds.
method Singular banded unlink diagrams and associated moves.
result Every immersed surface can be represented uniquely by a singular banded unlink diagram.
The paper shows how reducible complexes affect local indicability.
problem The local indicability of subcomplexes in reducible complexes.
method Characterization of diagrammatic reducibility and application to local indicability.
result Injective labeled oriented trees are locally indicable if reducible of degree 2.
Diagrammatic method calculates knot invariant related to Chern-Simons theory.
problem Calculating knot invariants in Chern-Simons theory.
method Diagrammatic approach for compact oriented 3-manifolds.
result Diagrammatic method for computing knot invariant .
Rewriting theory applied to diagrammatic algebras for categorification.
problem Finding bases in graded -foams.
method Algorithmic approach combining linear and higher rewriting, modulo rules capturing categorical properties.
result First proof of a basis theorem for graded -foams.
Pseudodiagrams are knot or link diagrams where some of the crossing information is missing. Pseudoknots are equivalence classes of pseudodiagrams, where equivalence is generated by a natural set of Reidemeister moves. In this paper, we introduce a Gauss-diagrammatic theory for pseudoknots which gives rise to the notion…
This paper uses sheaf theory to model virtual knots geometrically.
problem Defining and understanding virtual knots in a geometric framework.
method Sheaf theory applied to virtual knot diagrams to model them geometrically.
result A geometric model for virtual knots formalizes the intuitive notion of knots in a variable ambient space.
New theory classifies knotted spheres in 4D space.
problem Classifying knotted punctured spheres in 4D space.
method Diagrammatic theory of welded graphs, Tube map extension, Milnor invariants.
result Complete link-homotopy classification of knotted punctured spheres.
Survey of various non-classical knot theories from geometric and algebraic perspectives.
problem Various modifications to classical knot theory.
method Comparative geometric and algebraic analysis of non-classical knot theories.
result Distinct topological and combinatorial features in generalized knot theories.
Revives Vogel's diagrammatic technique for universal Lie algebra computations.
problem The universality of Lie algebra quantities remains open, despite many being described.
method Diagrammatic algebra based on Vogel's Λ-algebra.
result Diagrammatic technique enables truly universal computations in Lie theory.
This is an expository article on diagrammatic representations of knots and links in various settings via braids.
In this paper we study some aspects of knots and links in lens spaces. Namely, if we consider lens spaces as quotient of the unit ball with suitable identification of boundary points, then we can project the links on the equatorial disk of , obtaining a regular diagram for them. In this contest, we obtai…
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 …
Fixed point sets of certain group actions are contractible.
problem Fixed point sets of group actions on specific types of complexes.
method Analyzing group actions on diagrammatically reducible complexes with fine 1-skeleton.
result Fixed point sets are contractible under certain conditions.
Diagrammatic method calculates knot invariant from tangle decompositions.
problem Computing Rasmussen's invariant for knots.
method Diagrammatic approach using tangles and cobordisms.
result Computed -invariants of all 3-strand pretzel knots.
A diagrammatic definition of math.GT
We give a diagrammatic definition of when is not a root of unity, including its Hopf algebra structure and its relationship with the Temperley-Lieb category.
We define two functors from Elias and Khovanov's diagrammatic Soergel category, one targeting Clark-Morrison-Walker's category of disoriented sl(2) cobordisms and the other the category of (universal) sl(3) foams.
The article improves the display of acceptable exchange ratios for merging companies.
problem Determining feasible exchange ratios for merging companies.
method Exploits a diagrammatic approach to display the bargaining region.
result Shares face upper and lower bounds for acceptable exchange ratios.
This paper applies knot theory to modern yo-yo play.
problem No systematic knot-theoretic treatment of contemporary yo-yo play.
method Recalled fundamental knot theory results and developed a methodology for classifying string arrangements (mounts).
result Classified a range of mounts and identified corresponding yo-yo maneuvers.
We define and compare several natural ways to compute the bridge number of a knot diagram. We study bridge numbers of crossing number minimizing diagrams, as well as the behavior of diagrammatic bridge numbers under the connected sum operation. For each notion of diagrammatic bridge number considered, we find crossing …
Diagrammatic calculus proves Alexander polynomial formulas.
problem Proving formulas for Alexander polynomial.
method Diagrammatic calculus of quantum representations.
result Quantum invariant determines Alexander polynomial.
The random matrix theory method of planar Gaussian diagrammatic expansion is applied to find the mean spectral density of the Hermitian equal-time and non-Hermitian time-lagged cross-covariance estimators, firstly in the form of master equations for the most general multivariate Gaussian system, secondly for seven part…
We present a new test for studying asphericity and diagrammatic reducibility of group presentations. Our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails. We use this criterion to generalize results of J. Howie and S.M. Gersten on asphericity of LOTs and of Adian pre…
We provide a diagrammatic computation for the bilinear form, which is defined as the pairing between the (relative) cup products with every local coefficients and every integral homology 2-class of every links in the 3-sphere. As a corollary, we construct bilinear forms on the twisted Alexander modules of links.
The Ptolemy groupoid is a combinatorial groupoid generated by elementary moves on marked trivalent fatgraphs with three types of relations. Through the fatgraph decomposition of Teichmüller space, the Ptolemy groupoid is a mapping class group equivariant subgroupoid of the fundamental path groupoid of Teichmüller space…
Diagrammatic proof of twist theorem for 4-manifolds.
problem Understanding twists of 4-manifolds.
method Using trisection diagrams for purely diagrammatic proof.
result Answered a question about twists of 4-manifolds.
Develops a diagrammatic method for symplectic filling classifications.
problem Classifying exact/weak symplectic fillings of 3D contact manifolds.
method Symplectic JSJ decomposition applied to contact surgery diagrams.
result Recover symplectic fillings for certain lens spaces and torus bundles, and classify fillings for a large class of plumbed 3-manifolds.
New method for simplifying complex 4D shapes with boundaries.
problem Complexity reduction in 4D shape analysis.
method Introducing pseudo-trisections and pseudo-bridge trisections.
result Existence and uniqueness of pseudo-trisections established.
New diagrammatic approach connects knot Floer homology with quantum group representations.
problem Efficiently compute knot Floer homology using bordered algebra.
method Identify endomorphism algebras and use Sartori's diagrammatic formulation.
result Diagrammatic reinterpretation of Sartori's algebra and new modules over Ozsváth-Szabó's algebra.
New pseudometrics defined on knot spaces based on curve thickness and length.
problem Rigidity and non-degeneracy of knot spaces under isotopies.
method Swept-area pseudometrics on ropelength-filtered knot spaces.
result Proved non-degeneracy on polygonal strata and exact distance formulas.