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. 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. K. Ichihara and E. Matsudo introduced the notions of Z-colorable links and the minimal coloring number for Z-colorable links, which is one of invariants for links. They proved that the lower bound of minimal coloring number of a non-splittable Z-colorable link is 4. In this paper, we sh…
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. In this paper, we show that the minimal coloring number of any non-splittable Z-colorable links is four. As an example, we consider the link obtained by…
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 …
Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of links. By using the HOMFLY-PT skein theory, firstly, we show that the (reformulate…
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.
Extends link colorings to modules over Laurent polynomial rings, showing isomorphisms and dimensions.
problem Extending link colorings to modules over Laurent polynomial rings.
method Using Alexander quandles and modules over Laurent polynomial rings, showing isomorphisms and dimensions.
result Dimension of colorings as vector spaces over fields is determined by ring homomorphisms.
Polynomial invariant of quandles counts random link colorings.
problem Counting Q-colorings of random braids in quandles. method Average number of Q-colorings for large n. result The average number of Q-colorings coincides with a polynomial PQ. The Conway potential function (CPF) for colored links is a convenient version of the multi-variable Alexander-Conway polynomial. We give a skein characterization of CPF, much simpler than the one by Murakami. In particular, Conway's `smoothing of crossings' is not in the axioms. The proof uses a reduction scheme in a t…
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.…
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.
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. Computes link invariants in real projective 3-space using topological vertex.
problem Computing link invariants in RP3. method Uses geometric transition and toric Calabi-Yau threefold.
result Findings are series in Kahler parameters with positivity property.
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.
3D gauge theories link knot polynomials to vortex partition functions.
problem Connecting knot polynomials to gauge theory partition functions.
method Construct 3D N=2 abelian gauge theories on S2imesS1. result Colored Jones polynomials derived from vortex partition functions.
For the potential function of a link diagram induced by the optimistic limit of the colored Jones polynomial, we show the existence of a solution of the hyperbolicity equations by directly constructing it. This construction is based on the shadow-coloring of the conjugation quandle induced by a boundary-parabolic repre…
An invariant for knots with colored bonds respects HOMFLYPT relation.
problem Modeling and distinguishing knots with colored bonds.
method Introducing a HOMFLYPT skein module for colored bonded knots.
result The non-rigid version of the module provides information about knottedness of bonds.
Legendrian knot representations linked to colored Kauffman polynomial.
problem Relating Legendrian knot representations to colored Kauffman polynomial.
method Introducing ungraded n-colored ruling polynomial and relating it to the n-colored Kauffman polynomial. result Ungraded representation numbers of Legendrian knot DG-algebra agree with n-colored Kauffman polynomial specialization. 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.
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.
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.
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 …
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.
New deformation of link homology for colored diagrams.
problem Understanding colored Khovanov-Rozansky homology.
method Introducing a multi-parameter deformation and extending to braids.
result Link splitting properties and invariants of colored Hopf links.
The paper introduces two-tone colorings for links and shows conditions for surjective dihedral representations.
problem The challenge is to find conditions for links to admit surjective dihedral representations.
method The method involves introducing two-tone colorings and providing conditions for the link groups to admit such representations.
result Any link with at least 3 components admits a surjective homomorphism to the dihedral group of arbitrary degree.
Study homology of torus links and colored knots.
problem Computing homology of specific link types.
method Khovanov-Rozansky homology for positive torus links and colored knots.
result Computed homology for a family of links including torus links and colored knots.
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.
New link invariants from diagram colorings match link widths.
problem Defining link widths via diagram colorings.
method Colorings of link diagrams to define invariants and prove their equivalence to link widths.
result Invariants of link widths calculated algorithmically.
Invariants from biquasile colorings distinguish surface-links.
problem Counting and distinguishing surface-links.
method Coloring oriented surface-links using biquasiles and marked graph diagrams.
result Invariants can distinguish closed surface-links and cobordisms.
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.
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.
The paper introduces colorings and invariants for twisted links and shows how double coverings can be equivalent.
problem Understanding and distinguishing twisted links.
method Twisted intersection colorings and double coverings.
result There exist infinitely many pairs of twisted links with equivalent double coverings.
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 (…
In this paper, we use `generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in S^3. This yields an integral valued function on the m-dimensional torus, where m is the number of colors of the link. The case m=1 corresponds to the Levine-Tristram signature. We show that many remarkable p…
In-degree quiver polynomials for surface-links computed.
problem Computing in-degree quiver polynomials for surface-links.
method Defined using a quandle and set of endomorphisms, computed for surface-links with ch-index up to 10.
result Example computations for surface-links with ch-index up to 10.
We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further,…
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.
A quandle coloring obstruction prevents a specific link from being ribbon concordant.
problem Obstructing a specific link from being ribbon concordant.
method Symmetric dihedral quandle coloring analysis.
result A symmetric dihedral quandle of order 4 cannot color a specific surface-link, obstructing ribbon concordance.
The paper defines coloring invariants for links in a specific surface.
problem No specific problem stated; defining invariants for links.
method Defining coloring invariants for links in Σ_g × S^1.
result Definitions and generalizations of coloring invariants for links.
The paper studies polynomials and ideals from colored Jones polynomials for links.
problem Understanding the structure of colored Jones polynomials for links.
method Investigates commutative and noncommutative ideals derived from colored Jones polynomials.
result Formulates the link version of the AJ conjecture.
This paper establishes a correspondence between biquandle and quandle colorings for classical and surface links.
problem Finding refined invariants for classical and surface links using biquandles.
method Explicit one-to-one correspondence between biquandle colorings and quandle colorings.
result Biquandle homotopy invariants and quandle homotopy invariants are equivalent.
The colored Jones polynomial is a q-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A q-series called a tail is obtained as the limit of the sl2 colored Jones polynomials {Jn(K;q)}n for some link K, for example, an alternating link. For the $\mathf…
The paper connects Kirby diagrams and 5-colored graphs to represent 4-manifolds.
problem Representing compact 4-manifolds using Kirby diagrams and graphs.
method Algorithmically constructing 5-colored graphs from Kirby diagrams to represent PL 4-manifolds.
result Upper bounds for gem-complexity and regular genus derived from Kirby diagrams.
New method for calculating colored HOMFLY-PT polynomials for links with different symmetric representations.
problem Calculating colored HOMFLY-PT polynomials for links with arbitrary symmetric representations.
method Using quantum Racah coefficients (6j-symbols) of Uq(sl2) to simplify the evaluation. result Multi-colored link polynomials H[r1],[r2] for a specific link L7a3 are successfully evaluated. 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…
We prove that the coefficients of the colored Jones polynomial of alternating links stabilize under increasing the number of twists in the twist regions of the link diagram. This gives us an infinite family of q-power series derived from the colored Jones polynomial parametrized by the color and the twist regions of …