arXiv research

The contents of this 98-page paper have been subsumed into the 191-page paper "A colored sl(N)-homology for links in S^3" (arXiv:0907.0695v1 [math.GT]), in which we further develop the theory and use it to construct a colored link homology.

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.

Generalizes tangle Floer homology to A-tangles with A-colorings.

A new method computes link invariants from diagrams.

The evolution of prices on ideal market is given by geometrical Brownian motion, where Gaussian white noise describes fluctuations. We study the effect of correlations introduced by a color noise.

The slope of a colored link in an integral homology sphere is a rational function that generalizes the Kojima-Yamasaki η-function.

Budney recently constructed an operad that encodes splicing of knots. He further showed that the space of (long) knots is generated over this operad by the space of torus knots and hyperbolic knots, thus generalizing the satellite decomposition of knots from isotopy classes to the level of the space of knots. Infection…

Logarithmic invariant for restricted quantum sl(2) constructed.

Graph coloring is explained using a topological field theory with defects.

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…

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 (…

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…

The paper introduces a new coloring invariant for spatial surfaces using a multiple group rack.

This paper unifies observability notions for colored graphs and identifies graph modifications to improve observability.

If all but two vertices of a triangulated sphere have degrees divisible by $k$, then the exceptional vertices are not adjacent. This theorem is proved for $k=2$ with the help of the coloring monodromy. For $k = 3, 4, 5$ colorings by the vertices of platonic solids have to be used. With a coloring monodromy one can asso…

We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N) knot polynomial due to Murakami, Ohtsuki, and Yamada. We discuss how the resulting moduli space can be thought of a representation variety. We show that the Euler characteristic…

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

It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight system. We give two curious formulas for the weight system of a colored Jones fun…

Algorithm calculates ribbon obstructions for colored knots.

Taking the signature of the closure of a braid defines a map from the braid group to the integers. In 2005, Gambaudo and Ghys expressed the homomorphism defect of this map in terms of the Meyer cocycle and the Burau representation. In the present paper, we simultaneously extend this result in two directions, considerin…

A quandle coloring obstruction prevents a specific link from being ribbon concordant.

The study of tiling homology on flat surfaces, proving impossibility of certain tilings.

Fix an integer N>1. To each diagram of a link colored by 1,...,N, we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain com…

New invariant slope for links helps complete signature formula.

In this article we describe a canonical way to expand a certain kind of $(\mathbb Z_2)^{n+1}$-colored regular graphs into closed $n$-manifolds by adding cells determined by the edge-colorings inductively. We show that every closed combinatorial $n$-manifold can be obtained in this way. When $n\leq 3$, we give simple eq…

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,…

Hybrid deep learning algorithm optimizes register allocation for compiler.

We give a construction of quandle cocycles from group cocycles, especially, for any integer p \geq 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group Z/p. We will show that a group 3-cocycle of Z/p gives rise to a non-trivial quandle 3-cocycle of R_p. When p is an odd prime, since d…

We formulate a conjecture (already proven by A. Kricker) about the structure of Kontsevich integral of a knot. We describe its value in terms of the generating functions for the numbers of external edges attached to closed 3-valent diagrams. We conjecture that these functions are rational functions of the exponentials …

A stock market visualization tool using 'functional visualization' concept.

New algebraic structure for 2-string links and long knots.

A coloring scheme improves graph neural networks for node disambiguation.

Categorifies quantum invariants using cobordism categories and operads.

Vision impairment due to pathological damage of the retina can largely be prevented through periodic screening using fundus color imaging. However the challenge with large scale screening is the inability to exhaustively detect fine blood vessels crucial to disease diagnosis. In this work we present a computational ima…

This paper consists of three parts. First, we generalize the Jaeger Formula to express the Kauffman-Vogel graph polynomial as a state sum of the Murakami-Ohtsuki-Yamada graph polynomial. Then, we demonstrate that reversing the orientation and the color of a MOY graph along a simple circuit does not change the sl(N) Mur…

New colored knot Floer homology defined using infinite full twists.

Proves existence of colored Khovanov bicomplex linking Jones polynomial.

Defines a new version of Turaev-Viro invariants for 3-manifolds with boundaries.

Study uses knot theory to model RNA foldings, emphasizing both entanglement and intrachain interactions.

Probabilistic fair clustering tackles uncertain group membership.

We extend Lipshitz-Sarkar's definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. Given an assignment c (called a coloring) of positive integer to each component of a link L, we define a stable homotopy type X_col(L_c) whose cohomology recovers the c-colored…

New algorithms learn staged trees from incomplete data.

Asymptotically CAT(0) metrics and Z-structures for HHGs, proving Farrell-Jones Conjecture.

Quaternion CNN outperforms traditional CNN in color image reconstruction.

Finite type invariants separate PL links in 3D space.

Let N be a regular branched cover of a homology 3-sphere M with deck group G isomorphic to Z_2^d and branch set a trivalent graph Gamma; such a cover is determined by a coloring of the edges of Gamma with elements of G. For each index-2 subgroup H of G, M_H = N/H is a double branched cover of M. Sakuma has proved that …