Minimal coloring number found for Z-colorable links.
problem Finding the minimum number of colors needed for Z-colorings of Z-colorable links.
method Defined Z-coloring as a generalization of Fox coloring for links with zero determinants. Provided sufficient conditions for non-splittable Z-colorable links to have the least minimal coloring number.
result Sufficient conditions for non-splittable Z-colorable links to have the least minimal coloring number.
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. 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. 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. 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. Relations will be described between the quandle cocycle invariant and the minimal number of colors used for non-trivial Fox colorings of knots and links. In particular, a lower bound for the minimal number is given in terms of the quandle cocycle invariant.
We show that the minimal number of colors for all effective n-colorings of a link with non-zero determinant is at least 1+log2n.
A link diagram is said to be lune-free if, when viewed as a 4-regular plane graph it does not have multiple edges between any pair of nodes. We prove that any colored link diagram is equivalent to a colored lune-free diagram with the same number of colors. Thus any colored link diagram with a minimum number of colors (…
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.
In this paper we first investigate minimal sufficient sets of colors for p=11 and 13. For odd prime p and any p-colorable link L with non-zero determinant, we give alternative proofs of mincol_p L \geq 5 for p \geq 11 and mincol_p L \geq 6 for p \geq 17. We elaborate on equivalence classes of sets of distinct colors (o…
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…
The paper tackles fair correlation clustering with fairness constraints.
problem Minimizing disagreements while adhering to fairness constraints for clustering.
method Two variants of fairness constraints are considered: equal distribution and relative bounds. Approximation algorithms are developed for these constraints.
result Approximation algorithms for fair correlation clustering with theoretical guarantees and empirical validation.
The paper finds braid representatives minimizing simple walks for knots.
problem Finding efficient braid representatives for knots.
method Developed methods to minimize the number of simple walks in braids.
result Computed the colored Jones polynomial for specific knots.
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. If a knot has the Alexander polynomial not equal to 1, then it is linear n-colorable. By means of such a coloring, such a knot is given an upper bound for the minimal quandle order, i.e., the minimal order of a quandle with which the knot is quandle colorable. For twist knots, we study the minimal quandle orders in d…
Minimal complexes for two-strand braids defined directly.
problem Determining minimal complexes for braids.
method Explicit formulas derived from educated guesswork and reverse engineering.
result Construction of minimal complexes homotopy equivalent to Rickard complexes.
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.
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 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.
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.
For each prime p > 7 we obtain the expression for an upper bound on the minimum number of colors needed to non-trivially color T(2, p), the torus knots of type (2, p), modulo p. This expression is t + 2 l -1 where t and l are extracted from the prime p. It is obtained from iterating the so-called Teneva transformations…
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.
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 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.
The paper calculates minimum Dehn colors for knots using symmetric local biquandle cocycles.
problem Determining the minimum number of Dehn colors for knots.
method Using symmetric local biquandle cocycle invariants to evaluate minimum Dehn colors.
result There exist knots distinguished by minimum numbers of Dehn colors.
New method finds 198,846 toric-colorable seeds of Picard number 5.
problem Enumerating toric-colorable seeds of Picard number 5.
method Binary matroid approach and dynamic programming algorithm.
result 198,846 mod 2 toric-colorable seeds of dimension four and Picard number five.
The study of coloring points in hyperbolic plane and related discrete structures.
problem Coloring points in the hyperbolic plane to avoid same color for points at specific distance.
method Using a strategy similar to Kloeckner, the paper shows linear upper bounds on the number of colors needed.
result Linear upper bounds on the number of colors needed for coloring points in the hyperbolic plane and related discrete structures.
The paper studies unlinking links with minimal crossing changes.
problem Unlinking links with minimal changes when crossings between components are fixed.
method Examined minimum number of crossing changes for links with linking number zero and unknotted components.
result Data and general results about asymmetry of unlinking between components for links with up to ten crossings.
Explains a 2D color exchange invariant correspondence to 3D linking numbers.
problem Understanding color exchange invariants in 2D dynamics and their 3D geometric interpretation.
method Visualizes invariants as linking of lines on a special surface with Arf-Kervaire invariant one, and interprets it as an obstruction to continuous transformation.
result Interprets a 2D color exchange invariant as a 3D linking number, providing a topological explanation.
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.
The minimum number of colors is a challenging knot invariant since, by definition, its calculation requires taking the minimum over infinitely many minima. In this article we estimate and in some cases calculate the minimum number of colors for the Turk's head knots on three strands.
Lower bounds for handlebody-knots using Alexander biquandle colorings.
problem Calculating Gordian distance and unknotting number for handlebody-knots.
method Using Alexander biquandle colorings to construct and calculate handlebody-knots.
result Constructed handlebody-knots with specific distance and unknotting numbers.
Paper studies knotoid chirality using shadow quandle colorings and invariants.
problem Distinguishing knotoids from their mirrors.
method Shadow quandle colorings and cocycle invariants.
result Knotoid 31 is shown to be chiral. The dihedral genus of a knot is related to its signature and minimal surface genus.
problem Determining the dihedral genus of knots and its relation to signatures and minimal surface genus.
method Using dihedral branched covers and properties of Fox colorings, the dihedral genus is linked to the signature of the cover and the Murasugi signature of the knot.
result An infinite family of knots with minimal genus surfaces that realize the four-genus of the knot.
This article concerns exact results on the minimum number of colors of a Fox coloring over the integers modulo r, of a link with non-null determinant. Specifically, we prove that whenever the least prime divisor of the determinant of such a link and the modulus r is 2, 3, 5, or 7, then the minimum number of colors is 2…
The paper extends surface link coloring theory to triplane diagrams and knots.
problem Understanding the topological properties of knots and surfaces in 4-space.
method Translated Niebrzydowski's theory of region colorings to triplane diagrams and movies of knots, providing inequalities and applications.
result Yoshikawa's 2-knots 91 and 102 are non-invertible. We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of…
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…
New Arf invariants for colored links determined by linking numbers.
problem Extending Arf invariant to colored links.
method Using generalized Seifert forms to construct quadratic forms and determining Arf invariant.
result New Arf invariants for colored links are determined by linking numbers.
In this article we take up the calculation of the minimum number of colors needed to produce a non-trivial coloring of a knot. This is a knot invariant and we use the torus knots of type (2, n) as our case study. We calculate the minima in some cases. In other cases we estimate upper bounds for these minima leaning on …
New method calculates bridge indices of spatial graphs using diagram colorings and Wirtinger number.
problem Calculating bridge indices for spatial graphs efficiently.
method Extending Wirtinger number to spatial graphs, implementing Python algorithm, combining algebraic structures and clasping techniques.
result Exact bridge indices for almost unknotted graphs of large bridge index.
Study on coloring curves on surfaces, showing growth and unique colorability.
problem Chromatic number of curve graphs on surfaces.
method Analyzing separating curves and homology classes, using Kneser graphs and hyperbolic geometry.
result Chromatic number grows like k log k for separating curves, and uniquely t-colorable for homology classes.
Study compact PL 4-manifolds with special handle decompositions.
problem Existence of special handlebody decompositions for simply-connected closed PL 4-manifolds.
method Investigate colored triangulations inducing handle decompositions without 1-handles or 1- and 3-handles.
result Detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary that admit such decompositions.
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.
Coloring numbers are one of the simplest combinatorial invariants of knots and links to describe. And with Joyce's introduction of quandles, we can understand them more algebraically. But can we extend these invariants to tangles -- knots and links with free ends? Indeed we can, once we categorify. Starting from the de…
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.
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.
This article is about chromatic numbers of hyperbolic surfaces. For a metric space, the d-chromatic number is the minimum number of colors needed to color the points of the space so that any two points at distance d are of a different color. We prove upper bounds on the d-chromatic number of any hyperbolic surfac…