Study on knot diagrams showing bridge number can differ from crossing number.
problem Incompatibility between crossing number and bridge number for knot diagrams.
method Defined and compared various bridge number computations for knot diagrams, studied minimizing diagrams, and constructed families of minimal crossing diagrams.
result Found examples where bridge number differs from crossing number, and demonstrated this difference can grow infinitely.
Average signature of 2-bridge knots approximates sqrt(2c/π).
problem Estimating the average signature and 4-genus of 2-bridge knots.
method Developed a model for 2-bridge knot diagrams indexed by crossing number, and used it to derive upper bounds for the average 4-genus.
result Upper bound for the average 4-genus of a 2-bridge knot is 9.75c/log c.
We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of four-manifolds and extends the classical concept of bridge splittings of links in the three-sphere to four dimensions. We prove that every knotted surface in the four-spher…
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.
We prove every knotted surface can be isotoped to a bridge position in a 4-manifold.
problem Understanding knotted surfaces in 4-manifolds.
method Introducing generalized bridge trisections and shadow diagrams.
result Existence of exotic 4-manifolds with certain trisections.
We generalize bridge trisections to surfaces in four-manifolds, linking them to braided links.
problem Embedding surfaces in four-manifolds and understanding their braiding.
method Introducing bridge trisections, isotoping surfaces, and using trisections and open-book decompositions.
result Any neatly embedded surface can be isotoped to lie in bridge trisection position with respect to any trisection of the four-manifold.
The study proves knots and certain links support taut foliations.
problem Proving knots and links support taut foliations.
method Diagrammatic properties and weighted planar trees.
result Knots and certain links support coorientable taut foliations.
We discuss the fundamental (relative) 3-classes of knots (or hyperbolic links), and provide diagrammatic descriptions of the push-forwards with respect to every link-group representation. The point is an observation of a bridge between the relative group homology and quandle homology from the viewpoints of Inoue--Kabay…
Introduces directed diagrammatic reducibility with group and topological implications.
problem Diagrammatic reducibility in relative presentations.
method Adapting classical tools for diagrammatic reducibility to directed diagrammatic reducibility.
result Strong group theoretic and topological consequences of directed diagrammatic reducibility.
Classifies L-space surgeries on all two-bridge links
problem Classifying L-space surgeries on two-bridge links method Introduces a sufficient diagrammatic condition for links in S3 to be persistently foliar, defines a simplified model for Heegaard Floer homology, and uses Turaev torsions for computations result Determines L-space surgeries in the case of generalised L-space links 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.
Study of curves in rational surfaces using multisections and torus actions.
problem Understanding curves in rational surfaces using multisections and torus actions.
method Analysis of multisections of embedded surfaces in rational 4-manifolds with torus actions.
result Every smooth, complex curve in CP^1 × CP^1 can be put in efficient bridge position with respect to a genus one 4-section.
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.
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.
Explains how knots and links can be shown using braids.
problem None explicitly stated; focuses on representation methods.
method Diagrammatic representations of knots and links via braids.
result Explains how braids can represent knots and links.
For each integer k≥4 we describe diagrammatically a positively graded Koszul algebra Dk such that the category of finite dimensional Dk-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type Dk or Bk−1, constructible with respect…
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 d. 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.
Rewriting theory applied to diagrammatic algebras for categorification.
problem Finding bases in graded gl2-foams. method Algorithmic approach combining linear and higher rewriting, modulo rules capturing categorical properties.
result First proof of a basis theorem for graded gl2-foams. New method finds infinitely many surface knots with specific bridge numbers.
problem Finding numerical invariants for surface links.
method Colorings of surface links by keis to prove bridge number existence.
result Existence of infinitely many surface knots with bridge number n for n ≥ 4.
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.
Paper refines generating function for 2-bridge knot groups.
problem Determining the number of epimorphisms between 2-bridge knot groups.
method Refined generating function considering genus and crossing number.
result Improved formula for epimorphisms between 2-bridge knot groups.
The paper calculates bridge numbers for knots using machine learning.
problem Determining the bridge number for virtual knots with multiple definitions.
method Employed computational techniques and machine learning models to classify knots based on their bridge numbers.
result Demonstrated that the bridge number for virtual knots can differ significantly.
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 s-invariants of all 3-strand pretzel knots. We provide a new proof of the following results of H. Schubert: If K is a satellite knot with companion J and pattern L that lies in a solid torus T in which it has index k, then the bridge numbers satisfy the following: 1) The bridge number of K is greater than or equal to the product of k and the bridge number of J; …
We show that there are hyperbolic tunnel-number one knots with arbitrarily high bridge number and that "most" tunnel-number one knots are not one-bridge with respect to an unknotted torus. The proof relies on a connection between bridge number and a certain distance in the curve complex of a genus-two surface.
Paper describes a state sum formula for a graph coloring polynomial.
problem Counting n-face colorings of ribbon graphs for various n. method Combines topological quantum field theory and diagrammatic tensors.
result Describes a state sum formula for the total face color polynomial.
For any given number of crossings c, there exists a formula to determine the number of 2-bridge knots of c crossings, and indeed it is a simple matter to actually construct presentations of these knots. However, the determination of whether a given (prime) knot is a 2-bridge knot remains a nontrivial exercise, and …
This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants (A- or B-adequacy), we prove that the growth of the…
Normal distribution found for 2-bridge knots signatures.
problem Distribution of signatures for 2-bridge knots.
method Calculated average signature, introduced s(c,σ), used limit theorem. result Distribution of signatures approaches normal as knot complexity increases.
The study calculates the average genus of 2-bridge knots based on their crossing numbers.
problem Determining the average genus of 2-bridge knots with a given crossing number.
method Analytical approach focusing on the properties of 2-bridge knots.
result Obtained the oblique asymptote of the average genus as crossing numbers increase.
The paper refines the three-page index for links, proving a new bound and characterizing specific links.
problem Investigating the three-page index invariant for links and proving bounds.
method Constructing three-page presentations from reduced link diagrams via binding circles and contractible subcomplexes.
result Proves a new bound for the three-page index and characterizes links achieving equality.
We give a diagrammatic definition of Uq(sl2) when q is not a root of unity, including its Hopf algebra structure and its relationship with the Temperley-Lieb category.
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.
Study determines Δ-unknotting numbers for two-bridge knots.
problem Determining the minimum number of Δ-moves to simplify two-bridge knots. method Examined two-bridge knots of specific types and calculated their Δ-unknotting numbers. result Identified two-bridge knots with Δ-unknotting number equal to one. The study of 2-bridge knots reveals a linear average braid index as crossing number increases.
problem Understanding the distribution of braid indices in 2-bridge knots.
method Analyzing the asymptotic behavior of braid indices for fixed crossing numbers.
result The average braid index of 2-bridge knots of crossing number c is asymptotically $ rac{c}{3}+ rac{11}{9}$. The paper explores meridional ranks of knotted surfaces and welded knots, proving equalities and relationships.
problem Investigating the Meridional Rank Conjecture for knotted surfaces and welded knots.
method Constructing knots with specific properties, establishing equalities, and using Tube map.
result Established the equality of bridge number and meridional rank for certain knots and knotted spheres.
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.
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.
New model shows average genus of 2-bridge knots grows linearly with crossing number.
problem Understanding the growth of Seifert genus for 2-bridge knots.
method Billiard table model for 2-bridge knots.
result Average genus of a 2-bridge knot with crossing number c asymptotically approaches c/4 + 1/12.
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.
Study on bounds of knot untangling for specific types of knots.
problem Determining upper limits for knot untangling.
method Defined warping degree, examined diagrams combinatorially.
result Upper bounds for unknotting and region unknotting numbers.
The paper sets a lower bound on the crossing number of 2-bridge knots and answers a question about their epimorphism number.
problem Determining the epimorphism number of 2-bridge knots and answering a specific question posed by Suzuki.
method Using techniques related to continued fraction expansions and parsings of 2-bridge knots.
result Established a lower bound on the crossing number of 2-bridge knots and answered Suzuki's question about the epimorphism number.
We compute the genus zero bridge numbers and give lower bounds on the genus one bridge numbers for a large class of sufficiently generic hyperbolic twisted torus knots. As a result, the bridge spectra of these knots have two gaps which can be chosen to be arbitrarily large, providing the first known examples of hyperbo…
Diagrammatic calculus proves Alexander polynomial formulas.
problem Proving formulas for Alexander polynomial.
method Diagrammatic calculus of quantum sl2 representations. result Quantum invariant determines Alexander polynomial.
We present a practical algorithm to determine the minimal genus of non-orientable spanning surfaces for 2-bridge knots, called the crosscap numbers. We will exhibit a table of crosscap numbers of 2-bridge knots up to 12crossings (all 362 of them).
Identifies all 2-bridge links up to 11 crossings.
problem Classifying 2-bridge links up to a certain number of crossings.
method Enumerated all 2-bridge links up to 11 crossings and calculated splitting numbers.
result Identified all 2-bridge links up to 11 crossings and calculated splitting numbers.