arXiv research

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.

Aicardi's invariant $F(L)$ is extended to colored singular links using graphical calculus.

This paper shows the minimal coloring number for certain $\mathbb{Z}$-colorable links is four.

This paper proves the minimal coloring number for a specific type of link is exactly 4.

The paper explores minimal coloring numbers for $\mathbb{Z}$-colorable links.

Study finds minimal coloring numbers for torus links using rack colorings.

Classifies colored links and spatial graphs up to colored link-homotopy.

Link colorings linked to Goeritz matrix.

New colored link invariants using multi-quandles.

Study on colored Jones polynomial and link complements.

Paper detects checkerboard colorability of virtual links using odd writhe and arrow polynomial.

The study characterizes torus links' coloring quivers using dihedral quandles.

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.

Functoriality proved for colored link invariants.

New deformation of link homology for colored diagrams.

The paper introduces two-tone colorings for links and shows conditions for surjective dihedral representations.

Study homology of torus links and colored knots.

This paper shows all elements in the 3-colorable subgroup of Thompson's group give 3-colorable links.

New link invariants from diagram colorings match link widths.

Invariants from biquasile colorings distinguish surface-links.

Study shows colored Jones invariants limit to link volumes.

New Arf invariants for colored links determined by linking numbers.

The paper introduces colorings and invariants for twisted links and shows how double coverings can be equivalent.

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.

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.

The paper defines coloring invariants for links in a specific surface.

The paper studies polynomials and ideals from colored Jones polynomials for links.

This paper establishes a correspondence between biquandle and quandle colorings for classical and surface links.

We define a Khovanov homotopy type for $sl_2(\mathbb{C})$ 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 $n\rightarrow\infty$, generalizing the tail behavior of the colored Jones …

Computes Blanchfield pairing for colored links with non-zero Alexander polynomial.

New method for calculating colored HOMFLY-PT polynomials for links with different symmetric representations.

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 …

Researchers found a $q$-series identity for a specific knot using $\mathfrak{sl}_3$ representations.

New $p$-colorable subgroup derived from Thompson's group.

Symmetric quandles provide new insights into link colorings.

The paper confirms a conjecture and extends arrow polynomial to twisted links.

Algorithm calculates Seifert matrices for colored links.

We show that the minimal number of colors for all effective $n$-colorings of a link with non-zero determinant is at least $1+\log_2 n$.

Using the colored Kauffman skein relation, we study the highest and lowest $4n$ coefficients of the $n^{th}$ unreduced colored Jones polynomial of alternating links. This gives a natural extension of a result by Kauffman in regard with the Jones polynomial of alternating links and its highest and lowest coefficients. W…

The paper introduces new structures for colored HOMFLY-PT invariants using skein theory.

Explains a 2D color exchange invariant correspondence to 3D linking numbers.