arXiv research

New invariant for square-free integers derived from kei theory.

New method finds infinitely many surface knots with specific bridge numbers.

Murakami-Ohtsuki-Yamada introduced an evaluation of certain oriented planar trivalent graphs with colored edges. This evaluation plays a key role in the evaluation of the colored HOMFLY polynomial of a link in 3-space and its Khovanov-Rozansky categorification. Our goal is is to give a generating series formula for the…

We generalize the Morton-Franks-Williams inequality to the colored $\mathfrak{sl}(N)$ link homology defined in arXiv:0907.0695, which gives infinitely many new bounds for the braid index and the self linking number. A key ingredient of our proof is a composition product for the general MOY graph polynomial, which gener…

A coloring scheme improves graph neural networks for node disambiguation.

The G-degree of colored graphs is a key concept in the approach to Quantum Gravity via tensor models. The present paper studies the properties of the G-degree for the large class of graphs representing singular manifolds (including closed PL manifolds). In particular, the complete topological classification up to G-deg…

In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the bracket polynomial and then extracted from this polynomial in terms of its exponents,…

Dihedral linking invariant uses knot colorings to distinguish knots.

The differential expansion is one of the key structures reflecting group theory properties of colored knot polynomials, which also becomes an important tool for evaluation of non-trivial Racah matrices. This makes highly desirable its extension from knots to links, which, however, requires knowledge of the $6j$-symbols…

We construct an action of a polynomial ring on the colored sl(2) link homology of Cooper-Krushkal, over which this homology is finitely generated. We define a new, related link homology which is finite dimensional, extends to tangles, and categorifies a scalar-multiple of the sl(2) Reshetikhin-Turaev invariant. We expe…

The paper shows links can be colored with fewer colors than previously thought.

Improved algorithm for low-discrepancy colorings with practical time complexity.

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.…

VCML learns concepts and metaconcepts from images and questions.

Study on knots using 17 colors, finding specific color assignments.

It was shown that any $\mathbb{Z}$-colorable link has a diagram which admits a non-trivial $\mathbb{Z}$-coloring with at most four colors. In this paper, we consider minimal numbers of colors for non-trivial $\mathbb{Z}$-colorings on minimal diagrams of $\mathbb{Z}$-colorable links. We show, for any positive integer $N…

Study identifies key metrics for small and large tick assets in LOBs.

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

The paper finds minimum Dehn colors for knots and defines useful graphs for coloring.

We determine the minimal number of colors for non-trivial $\mathbb{Z}$-colorings on the standard minimal diagrams of $\mathbb{Z}$-colorable torus links. Also included are complete classifications of such $\mathbb{Z}$-colorings and of such $\mathbb{Z}$-colorings by only four colors, which are shown by using rack colorin…

K. Ichihara and E. Matsudo introduced the notions of $\mathbb{Z}$-colorable links and the minimal coloring number for $\mathbb{Z}$-colorable links, which is one of invariants for links. They proved that the lower bound of minimal coloring number of a non-splittable $\mathbb{Z}$-colorable link is 4. In this paper, we sh…

Factor complexity $b_φ(n)$ for a vertex coloring $φ$ of a regular tree is the number of colored $n$-balls up to color-preserving automorphisms. Sturmian colorings are colorings of minimal unbounded factor complexity $b_φ(n) = n+2$. In this article, we prove an induction algorithm for Sturmian colorings using colored ba…

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.

A strong interaction is known to exist between edge-colored graphs (which encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (as a possible approach to the study of Quantum Gravity). The key tool is the {\it G-degree} of the involved graphs, which drives the {\it $1/N$ expansion} in the tensor …

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.

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.

The minimal coloring number of a $\mathbb{Z}$-colorable link is the minimal number of colors for non-trivial $\mathbb{Z}$-colorings on diagrams of the link. In this paper, we show that the minimal coloring number of any non-splittable $\mathbb{Z}$-colorable links is four. As an example, we consider the link obtained by…

New invariant for virtual links defined using homology.

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.

Study of quandle coloring quivers with 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.

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.

The paper finds 3-colorings of 2-sphere triangulations.

Study on quandle coloring quivers for (p, 2)-torus knots and links.

New colored link invariants using multi-quandles.

Paper extends Enami-Ozeki-Yamaguchi's work on planar quadrangulations.

Gradient descent with error feedback performs better than vanilla when features are rare.

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

New colored knot Floer homology defined using infinite full twists.

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 …

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.