The paper studies quasi-Sturmian colorings on regular trees, distinguishing bounded and unbounded types.
problem Coloring regular trees with quasi-Sturmian properties.
method Developed an induction algorithm similar to Sturmian colorings, distinguishing types by recurrence function.
result Obtained an induction algorithm for quasi-Sturmian colorings on regular trees.
Algorithm finds minimal colorings of tree structures.
problem Finding minimal unbounded factor complexity colorings of trees.
method Induction algorithm using colored balls.
result Characterization of Sturmian colorings.
Study trajectories on homothety surfaces, linking to square torus and revealing mixed behaviors.
problem Analyzing linear trajectories on homothety surfaces.
method Examined a 1-parameter family of genus-2 homothety surfaces, considering their linear trajectories and their relation to square torus.
result Trajectories on homothety surfaces can contain either a closed loop or a lamination with Cantor cross-section, and their cutting sequences are either periodic or Sturmian.
In order to study large variations or fluctuations of finite or infinite sequences (time series), we bring to light an 1868 paper of Crofton and the (Cauchy-)Crofton theorem. After surveying occurrences of this result in the literature, we introduce the inconstancy of a sequence and we show why it seems more pertinent …
The paper explores minimal coloring numbers for Z-colorable links.
problem Finding the minimum number of colors needed for Z-colorings on minimal diagrams of Z-colorable links. method Investigates minimal diagrams and Z-colorings for Z-colorable links. result For any positive integer N, there exists a minimal diagram of a Z-colorable link with at least N colors in any Z-coloring. The paper shows links can be colored with fewer colors than previously thought.
problem Coloring links using the symmetric group of degree three.
method Analyzing the number of colors for link colorings by S3. result 2-bridge links with 5 colors can be colored with only 4 colors.
Study finds minimal coloring numbers for torus links using rack colorings.
problem Determining the minimal number of colors for torus link diagrams.
method Using rack colorings on link diagrams to classify minimal colorings.
result Complete classifications of Z-colorings by four colors. For a link with zero determinants, a Z-coloring is defined as a generalization of Fox coloring. We call a link having a diagram which admits a non-trivial Z-coloring a Z-colorable link. The minimal coloring number of a Z-colorable link is the minimal number of colors for non-trivial Z-colorings on diagrams of the link.…
Study on knots using 17 colors, finding specific color assignments.
problem Understanding the minimum number of colors needed for Fox colorings of knots.
method Investigated 17-colorable knots and their diagrams.
result Found that exactly 6 out of 17 colors are used in diagrams of 17-colorable knots.
Aicardi's invariant F(L) is extended to colored singular links using graphical calculus.
problem Constructing an invariant for colored classical and singular links.
method State-sum model using graphical calculus for oriented, colored, 4-valent planar graphs.
result Extends F(L) to colored singular links, showing it's stronger than HOMFLY-PT polynomial. This paper proves the minimal coloring number for a specific type of link is exactly 4.
problem Determining the minimal coloring number for a specific type of link.
method Investigated Z-colorable links and used their properties to prove the minimal coloring number is 4. result The minimal coloring number of any non-splittable Z-colorable link is exactly 4. The paper finds minimum Dehn colors for knots and defines useful graphs for coloring.
problem Finding the minimum number of colors for Dehn colorings of knots.
method Analyzes Dehn colorings for knots and defines R-palette graphs. result For Dehn p-colorable knots, the minimum number of colors is at least ⌊log2pfloor+2. This paper shows the minimal coloring number for certain Z-colorable links is four.
problem Determining the minimal number of colors for Z-colorings of links. method Analyzing diagrams of Z-colorable links and constructing specific diagrams to find the minimal coloring number. result The minimal coloring number for non-splittable Z-colorable links is four. We prove that any 11-colorable knot is presented by an 11-colored diagram where exactly five colors of eleven are assigned to the arcs. The number five is the minimum for all non-trivially 11-colored diagrams of the knot. We also prove a similar result for any 11-colorable ribbon 2-knot.
This survey article discusses three aspects of knot colorings. Fox colorings are assignments of labels to arcs, Dehn colorings are assignments of labels to regions, and Alexander-Briggs colorings assign labels to vertices. The labels are found among the integers modulo n. The choice of n depends upon the knot. Each typ…
The study characterizes torus links' coloring quivers using dihedral quandles.
problem Characterizing the structure of coloring quivers for torus links.
method Exhaustively determining all possible numbers of colorings and their interconnections.
result The quiver structure varies based on the number of colorings.
The paper discusses colorings and doubled colorings of virtual doodles.
problem Coloring virtual doodles using a new algebraic structure.
method Introduced a new algebra called a doodle switch and defined an invariant for virtual doodles.
result Introduced doubled colorings and defined an invariant for virtual doodles.
For each odd prime p, and for each non-split link admitting non-trivial p-colorings, we prove that the maximum number of Fox colors is p. We also prove that we can assemble a non-trivial p-coloring with any number of colors, from the minimum to the maximum number of colors. Furthermore, for any rational link, we prove …
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.
The paper discusses knot colorings and their invariants using Goeritz matrices.
problem Distinguishing knots using coloring methods.
method Elementary approach to equivalence between coloring and Goeritz matrices.
result Computing knot determinant and nullity of pretzel knots.
For any link and for any modulus m we introduce an equivalence relation on the set of non-trivial m-colorings of the link (an m-coloring has values in Z/mZ). Given a diagram of the link, the equivalence class of a non-trivial m-coloring is formed by each assignment of colors to the arcs of the diagram that is obtaine…
Introduced coloring-allowed invariants of planar knotoids with the coloring number.
problem Construction of polynomial invariants of knotoids with signs of crossings.
method Defined coloring-allowed invariants of planar knotoids with the coloring number.
result Discussed the 4-phases functions of coloring-allowed invariants.
Study of quandle coloring quivers with dihedral quandles.
problem Link invariants and their enhancements using quandles.
method Introduced shadow quandle coloring quivers and cocycle quivers, studied equivalence with quandle coloring numbers and shadow quandle cocycle invariants.
result Equivalence of quandle coloring quivers with quandle coloring numbers and shadow quandle cocycle quivers with shadow quandle cocycle invariants for specific dihedral quandles.
In this article we present the following new fact for prime p=11. For knots 6_2 and 7_2, mincol_{11} 6_2 = 5 = mincol_{11} 7_2, along with the following feature. There is a pair of diagrams, one for 6_2 and the other one for 7_2, each of them admitting only non-trivial 11-colorings using 5 colors, but neither of them a…
New TQFT homologies help color graphs, potentially solving the four color theorem.
problem Graph coloring problem, especially the four color theorem.
method Topological quantum field theory (TQFT) to define homology theories.
result TQFT homologies can generate 4-face colorings of bridgeless planar graphs, offering a constructive approach to the four color theorem.
Link colorings linked to Goeritz matrix.
problem Understanding link colorings and their relation to the Goeritz matrix.
method Exploring the relationship between link colorings and the Goeritz matrix.
result Established a connection between link colorings and the Goeritz matrix.
The ability to characterize the color content of natural imagery is an important application of image processing. The pixel by pixel coloring of images may be viewed naturally as points in color space, and the inherent structure and distribution of these points affords a quantization, through clustering, of the color i…
In this article we show that if a knot diagram admits a non-trivial coloring modulo 13 then there is an equivalent diagram which can be colored with 5 colors. Leaning on known results, this implies that the minimum number of colors modulo 13 is 5.
Study shows colored Jones invariants limit to link volumes.
problem Volume conjecture for colored Jones invariants.
method Deformation of hyperbolic structure for link complements.
result Limits of colored Jones invariants related to link volumes.
The paper finds 3-colorings of 2-sphere triangulations.
problem Coloring edges of triangulations of a 2-sphere in three colors.
method Enumerating triangulations and finding colorings by adding vertices.
result Other triangulations with less than 8 vertices have one unique coloring.
Gradient descent with error feedback performs better than vanilla when features are rare.
problem Improving communication complexity in distributed optimization with rare features.
method Gradient descent with greedy sparsification and error feedback for rare features.
result Communication complexity improves as features become more rare, potentially better than vanilla GD.
Study on quandle coloring quivers for (p, 2)-torus knots and links.
problem Understanding quandle colorings of (p, 2)-torus knots and links.
method Introduced quandle coloring quivers and studied them for dihedral quandles.
result Characterized quandle coloring quivers for (p, 2)-torus knots and links.
Paper extends Enami-Ozeki-Yamaguchi's work on planar quadrangulations.
problem Finding the maximum number of colors for proper anti-rainbow colorings on planar quadrangulations.
method Introducing half-monochromatic colorings for plane graphs with even polygonal faces and providing an upper bound in terms of the independence number.
result An upper bound on the maximum number of colors for half-monochromatic colorings is given in terms of the independence number.
Classifies colored links and spatial graphs up to colored link-homotopy.
problem Classifying colored links and spatial graphs up to colored link-homotopy.
method Using Habegger-Lin theory for colored string links, and extending to colored links and spatial graphs.
result Classification of colored links and spatial graphs up to colored link-homotopy.
New colored link invariants using multi-quandles.
problem Developing new invariants for colored links.
method Introducing multi-quandles and topological multi-quandles.
result New colored link invariants created.
New colored knot Floer homology defined using infinite full twists.
problem Defining a new homology theory for knots.
method Defining colored knot Floer homology through colimit of link Floer homology with infinite full twists.
result Colored knot Floer homology is a module over the colored knot Floer homology of the unknot.
We define a Khovanov homotopy type for sl2(C) colored links and quantum spin networks and derive some of its basic properties. In the case of n-colored B-adequate links, we show a stabilization of the homotopy types as the coloring n→∞, generalizing the tail behavior of the colored Jones …
We introduce three spectral sequences which give some expressions of colored Jones polynomials. Each spectral sequence contains a Khovanov-type homology groups. Two of them are derived from a bicomplex of the colored Jones polynomial. The other is the spectral sequence that deduces a colored Rasmussen invariant of link…
New knot homology invariant grows exponentially with color.
problem Constructing and understanding colored torus knot homology.
method Invariant construction and recursive formula for reduced HOMFLY homology.
result Doubly-graded invariant of positive torus knots grows exponentially in color.
A new method solves graph coloring problems using gradient descent.
problem Graph coloring problems, computationally difficult.
method Population-based weight learning framework, gradient descent on GPUs.
result Improved best-known results for several large graphs.
We show that the edges of every 3-connected planar graph except K4 can be colored with two colors in such a way that the graph has no color preserving automorphisms. Also, we characterize all graphs which have the property that their edges can be 2-colored so that no matter how the graph is embedded in any orienta…
This paper discusses reformulations of the problem of coloring plane maps with four colors. We give a number of alternate ways to formulate the coloring problem including a tautological expansion similar to the Penrose Bracket, and an extension of the Penrose Bracket that counts colorings of arbitrary cubic graphs pres…
The paper establishes a correspondence between quandle and biquandle colorings.
problem Understanding the relationship between quandle and biquandle colorings.
method Defining a functor and showing a one-to-one correspondence between colorings.
result The set of Alexander biquandles colorings is isomorphic to that of Alexander quandles colorings.
This paper shows all elements in the 3-colorable subgroup of Thompson's group give 3-colorable links.
problem Exploring the relationship between elements in the 3-colorable subgroup of Thompson's group and 3-colorable links.
method Defined the 3-colorable subgroup and used Jones's method to construct knots and links from elements of Thompson's group.
result All elements in the 3-colorable subgroup give 3-colorable links.
Study on colored Jones polynomial and link complements.
problem Understanding the structure of link complements with arbitrary colors.
method Investigated the potential function of the colored Jones polynomial and established a relationship with hyperbolicity.
result Evidence supports the Chen-Yang conjecture on link complements.
We prove that if a link admits non-trivial (2k+1)-colorings, with prime 2k+1>7, it also admits non-trivial (2k+1)-colorings not involving colors 2k, 2k-1, nor k.
Hybrid deep learning algorithm optimizes register allocation for compiler.
problem Efficiently coloring interference graphs for register allocation.
method Deep learning network trained on random graphs, augmented with a color correction phase.
result Hybrid algorithm performs well compared to optimal and greedy register allocators.
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.