Paper detects checkerboard colorability of virtual links using odd writhe and arrow polynomial.
problem Detecting checkerboard colorability of virtual links.
method Using odd writhe and arrow polynomial.
result Proves 6 virtual knots are not checkerboard colorable.
New polynomial for checkerboard-colorable 4-valent virtual graphs.
problem No specific problem stated; focuses on a new polynomial.
method Euler circuit expansion to assign polynomial to graphs.
result New combinatorial formulation of Kauffman-Jones polynomial.
The paper confirms a conjecture and extends arrow polynomial to twisted links.
problem Extending classical invariants to virtual and twisted links.
method Checkerboard framings, cut points, and normalized arrow polynomial.
result The normalized arrow polynomial is an invariant for twisted links.
Open books link prime positive braids to checkerboard graphs.
problem Determining link types of prime positive braids.
method Associated open books with checkerboard graphs.
result Link type of a prime positive braid closure is determined by the linking graph.
Proves certain alternating links have specific geometric properties.
problem Characterizing alternating links with totally geodesic checkerboard surfaces.
method Analyzes links with two totally geodesic checkerboard surfaces and characterizes them.
result Proves links with two totally geodesic checkerboard surfaces are three specific links.
A new copula, the checkerboard copula, maximizes entropy and preserves dependence.
problem Choosing copula for non-continuous marginal distributions.
method Introducing the checkerboard copula, maximizing Shannon entropy.
result Checkerboard copula maximizes entropy and preserves dependence.
Classifies fibering of state surfaces for various knot families.
problem Determining which state surfaces are fibered.
method Algebraic characterization of fibers from state graphs, decomposing graphs into planar components.
result Characterizes fibering for many families of state surfaces.
A method is presented to integrate explicitly a class of checkerboard IC-nets.
problem Integrating a class of checkerboard IC-nets.
method Based on Laguerre geometry and underlying pencils of conics and quadrics.
result Explicit parametrisation of checkerboard IC-nets.
Proposes a fixed smooth convolutional layer to reduce checkerboard artifacts in CNNs.
problem Checkerboard artifacts in CNNs during upsampling and strided convolution.
method Fixed convolutional layer with adjustable smoothness, applied to four CNNs and GANs.
result Significantly improves classification performance and image generation quality.
Geometric duality connects graph isomorphism and knot equivalence.
problem Understanding the equivalence of graph isomorphism and knot equivalence.
method Observation of geometric duality in planar graphs and links.
result The equivalence relation defined by isomorphisms of checkerboard graphs is the same as 2-isomorphisms of checkerboard graphs.
Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement stabilizes (approaches a geometric limit), but a corresponding checkerboard surface in…
Graphs with certain eigenvalues are linked to simply laced Dynkin diagrams.
problem Characterizing graphs with specific eigenvalues.
method Defining equivalence relation on signed graphs and using congruence over Z.
result Signed graphs with eigenvalues > -2 are linked to simply laced Dynkin diagrams.
NTK reveals order and chaos in DNNs, affecting checkerboard and border artifacts.
problem Checkerboard and border artifacts in DNNs.
method Analysis using Neural Tangent Kernel (NTK) in infinite-width setting.
result Transition between order and chaos regimes affects DNN performance.
The notion of chckerboard colorability for virtual links and abstract links is introduced. We study the Jones polynomials of virtual links and abstruct links. It is proved that a certain property of the Jones polynomials of classical links is valid for virtual links which admit checkerboard colorings.
The paper explores discrete isothermic nets using checkerboard patterns in quadrilateral nets.
problem Defining and understanding discrete isothermic nets in quadrilateral nets.
method Using checkerboard patterns and discrete differential geometry to define and analyze isothermic nets.
result The class of isothermic nets is invariant under dualization and Moebius transformations.
The paper studies right-angled links on higher genus surfaces.
problem Classifying and understanding right-angled links on surfaces of higher genus.
method Defining and proving equivalence of properties for RGCR links, using diagram restrictions and polygonal checkerboard surfaces.
result Classification of RGCR links and bounds on their number for a given genus.
IC-nets on confocal conics found in grid-like straight lines.
problem Understanding the geometric properties of grid-like straight lines with incircle quadrilaterals.
method Investigation of checkerboard IC-nets in plane and higher dimensions, using Laguerre geometry and 9 inspheres incidence theorem.
result Vertices of checkerboard IC-nets lie on confocal conics.
PixelDCL addresses checkerboard problem in deconvolutional layers.
problem Checkerboard problem in deconvolutional layers.
method Fresh interpretation of deconvolution operation to establish direct relationships among adjacent pixels.
result PixelDCL can consider spatial features and yields more accurate segmentation outputs.
Determinant modulo 8 classifies virtual knots based on polynomial coefficients.
problem Classifying virtual knots using determinant modulo 8.
method Introduced a determinant for checkerboard colorable virtual knots and proved its classification by the coefficient of z2 in the ascending polynomial. result Determinant modulo 8 classifies virtual knots based on polynomial coefficients.
The article establishes a polynomial for signed cyclic graphs and links it to checkerboard colorability.
problem Understanding graphical virtual links and their properties.
method Constructing virtual links from signed cyclic graphs, proving checkerboard colorability, and introducing a polynomial F[G].
result A virtual link is graphical if and only if it is checkerboard colorable.
Extends Gordon-Litherland pairing to links in thickened surfaces, defining new invariants.
problem Defining invariants for links in thickened surfaces.
method Extending Gordon-Litherland pairing, defining new invariants based on spanning surfaces.
result Invariants depend only on S∗-equivalence class of spanning surfaces and give well-defined invariants of virtual links. Essential surfaces in link diagrams on surfaces are crucial for understanding link properties.
problem Understanding the essentiality of surfaces in link diagrams on surfaces.
method Proving checkerboard surfaces are π1-essential and contain no essential closed curves that are ∂-parallel.
result Checkerboard surfaces of alternating link diagrams are π1-essential and contain no essential closed curves that are ∂-parallel.
Method converts virtual link diagrams to normal form, preserving equivalence.
problem Normalizing virtual link diagrams to study their properties.
method Method converts virtual link diagrams to normal form.
result Normal virtual link diagrams from equivalent diagrams are equivalent.
New invariant for virtual links defined using homology.
problem Invariants for virtual links and their properties.
method Defining homological arrow polynomial and using it to study virtual links.
result New invariant for virtual links and its properties.
Characterizes arithmetic and commensurable links in curved surfaces.
problem Classifying arithmetic and commensurable links in curved surfaces.
method Combines symmetry arguments, combinatorial geometry, and number-theoretic data.
result Characterizes arithmetic and commensurable right-angled tiling links.
It is shown that there exist alternating non-Montesinos knots whose essential spanning surfaces with maximal and minimal boundary slopes are not realised by the checkerboard surfaces coming from a reduced alternating planar diagram.
Developed neural network for predicting mechanical properties of composite materials.
problem Predicting and optimizing mechanical properties of composite materials.
method Convolutional neural network model integrated with a genetic algorithm optimizer.
result Highly accurate predictions and optimal microstructural designs identified.
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…
In this paper we review the definitions of homogeneous and alternative links. We also give two new characterizations of an alternative link diagram, one within the context of the enhanced checkerboard graph and another from the labeled Seifert graph.
New spanning tree model connects knot homology, s-invariant, and exotic discs.
problem Understanding exotic discs in the 4-ball for knots.
method Explicitly defined differential in spanning tree complex, described Rasmussen's s-invariant.
result Identified new infinite family of knots bounding exotic discs.
Improved linear upper bound for ribbonlength of knots.
problem Estimating the ribbonlength of knots and links.
method Using four-page open book decompositions and spanning trees of checkerboard graphs, constructing a four-page presentation with at most 2c(K) arcs.
result Proved that ribbonlength is bounded above by the four-page index, leading to the linear bound Rib(K) ≤ 2c(K).
Study of Khovanov homology for alternating virtual links, showing it's supported on specific diagonal lines.
problem Investigating the Khovanov homology of alternating virtual links.
method Analyzing the Khovanov homology of alternating virtual links and showing it's supported on specific diagonal lines.
result Khovanov homology of alternating virtual links is supported on g+2 diagonal lines, where g is the virtual genus. We show that the Kauffman bracket [L] of a checkerboard colorable virtual link L is an evaluation of the Bollobás-Riordan polynomial RGL of a ribbon graph associated with L. This result generalizes Thistlethwaite's celebrated theorem relating the Kauffman bracket with the Tutte polynomial of planar graphs.
A group-theoretical method, via Wada's representations, is presented to distinguish Kishino's virtual knot from the unknot. Biquandles are constructed for any group using Wada's braid group representations. Cocycle invariants for these biquandles are studied. These invariants are applied to show the non-existence of Al…
Parallelization technique for welded links preserves equivalence and yields specific decompositions.
problem Defining and proving equivalence of parallel welded link diagrams.
method Introduced a parallelization construction for welded link diagrams and showed its well-definedness.
result Parallel diagrams maintain equivalence under specific orientations and yield decompositions.
We introduce a topological combinatorial game called the Link Smoothing Game. The game is played on the shadow of a link diagram and legal moves consist of smoothing precrossings. One player's goal is to keep the diagram connected while the other player's goal is to disconnect the shadow. We make significant progress t…
We study the head and tail of the colored Jones polynomial while focusing mainly on alternating links. Various ways to compute the colored Jones polynomial for a given link give rise to combinatorial identities for those power series. We further show that the head and tail functions only depend on the reduced checkerbo…
Every link diagram can be represented as a signed ribbon graph. However, different link diagrams can be represented by the same ribbon graphs. We determine how checkerboard colourable diagrams of links in real projective space, and virtual link diagrams, that are represented by the same ribbon graphs are related to eac…
It is conjectured that the Khovanov homology of a knot is invariant under mutation. In this paper, we review the spanning tree complex for Khovanov homology, and reformulate this conjecture using a matroid obtained from the Tait graph (checkerboard graph) G of a knot diagram K. The spanning trees of G provide a filtrat…
The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists a complex generated by these spanning trees whose homology is the reduced Khovanov homology. The spanning trees provide a filtration on the reduced Khovanov complex and a …
The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the Tutte polynomial of planar graphs to graphs that are embedded in closed oriented s…
Method converts virtual link diagrams to normal ones.
problem Convert virtual link diagrams to normal ones.
method Double covering technique and generalized Reidemeister moves, Kauffman flypes.
result Normal virtual link diagrams obtained from equivalent virtual link diagrams are related by moves.
A classical result states that the determinant of an alternating link is equal to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link. We generalize this result to show that the determinant is the alternating sum of the number of quasi-trees of genus j of the dessin o…
We give constructions to realize an odd number, which is representable as sum of two squares, as determinant of an achiral knot, thus proving that these are exactly the numbers occurring as such determinants. Later we study which numbers occur as determinants of prime alternating achiral knots, and obtain a complete re…
The paper connects knot invariants to Laplacian matrices.
problem Computing knot invariants for various link diagrams.
method Using Laplacian matrices of directed, edge-weighted graphs derived from link diagrams.
result Principal minors of Laplacian matrices yield Seifert, Alexander, Goeritz matrices.
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We show that for any link diagram L, there is an associated ribbon graph whose quasi-trees correspond bijectively to …
Algorithm calculates Jones polynomial from Goeritz matrix.
problem Calculating Jones polynomial from link diagrams.
method Explicit algorithm using Goeritz matrices.
result Jones polynomial can be recovered from orientable checkerboard surfaces.
New invariant distinguishes all knots up to 10 crossings.
problem Classical knot invariants struggle with distinguishing all knots up to 10 crossings.
method Introducing a pair of integer polynomials associated with checkerboard planar graphs of minimal diagrams.
result The invariant distinguishes all knots up to 10 crossings.