arXiv research

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

This paper has two-fold goal: it provides gentle introduction to Knot Theory starting from 3-coloring, the concept introduced by R. Fox to allow undergraduate students to see that the trefoil knot is non-trivial, and ending with statistical mechanics. On the way we prove various (old and new) facts about knots. We rela…

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 $\mathfrak{sl}_2$ colored Jones polynomials $\{J_n(K;q)\}_n$ for some link $K$, for example, an alternating link. For the $\mathf…

Rosso and Jones gave a formula for the colored Jones polynomial of a torus knot, colored by an irreducible representation of a simple Lie algebra. The Rosso-Jones formula involves a plethysm function, unknown in general. We provide an explicit formula for the second plethysm of an arbitrary representation of $\fsl_3$, …

Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk with marked points. We derive some formulas for $A_1$ and $A_2$ clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formula…

This paper is base on talks which I gave in May, 2010 at Workshop in Trieste (ICTP). In the first part we present an introduction to knots and knot theory from an historical perspective, starting from Summerian knots and ending on Fox 3-coloring. We show also a relation between 3-colorings and the Jones polynomial. In …

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

Researchers compute and predict knot volumes using colored Jones polynomials.

Generates hard graph instances for algorithm analysis and benchmarking.

This paper is a next step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. In this paper, we managed to explicitly find the $\textit{inclusive}$ Racah matrices, i.e. the whole set of mixing matrices in channels $R^{\otimes 3}\longrightarrow Q$ with all possible $Q$, for …

The paper extends graph signatures to Klein graphs and foams, linking signatures to knot properties.

The Penrose-Kauffman polynomial connects knot theory to graph coloring.

The SL_3 colored Jones polynomial of the trefoil knot is a $q$-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this $q$-holonomic sequence as a case study. On the one hand, our results are new and useful to q…

The study characterizes maximal subgroups of a Thompson group and bounds the change in a knot invariant.

The Kauffman-Vogel polynomials are three variable polynomial invariants of $4$-valent rigid vertex graphs. A one-variable specialization of the Kauffman-Vogel polynomials for unoriented $4$-valent rigid vertex graphs was given by using the Kauffman bracket and the Jones-Wenzl idempotent colored with $2$. Bataineh, Elha…

Jones-Wenzl projectors lifted to Khovanov spectra, proving knot conjectures.

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

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

A small cover was introduced by Davis and Januszkiewicz as an $n$-dimensional closed manifold with a locally standard $Z_2)^n$-action such that its orbit space is a simple convex polytope. There exist a one-to-one correspondence between small covers and $(Z_2)^n$-colored polytopes. In this paper we study a construction…

Let $K\subset S^3$ be a Fox $p$-colored knot and assume $K$ bounds a locally flat surface $S\subset B^4$ over which the given $p$-coloring extends. This coloring of $S$ induces a dihedral branched cover $X\to S^4$. Its branching set is a closed surface embedded in $S^4$ locally flatly away from one singularity whose li…

Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander module…

Researchers calculate colored Jones polynomials for pretzel links using Kuperberg's theory.

The paper calculates colored Jones polynomials for specific link configurations.

Every cubic graph is a bridge trisection's 1-skeleton for a knotted surface.

Develops a method for random manifolds and submanifolds, focusing on 3-ball knots.

The Witten-Reshetikhin-Turaev invariants extend the Jones polynomials of links in S^3 to invariants of links in 3-manifolds. Similarly, in a preceding paper, the authors constructed two 3-manifold invariants N_r and N^0_r which extend the Akutsu-Deguchi-Ohtsuki invariant of links in S^3 colored by complex numbers to li…

Graphs on surfaces have a 2-dimensional large scale structure.

This paper is an extended account of my "Introductory Plenary talk at Knots in Hellas 2016" conference We start from the short introduction to Knot Theory from the historical perspective, starting from Heraclas text (the first century AD), mentioning R.Llull (1232-1315), A.Kircher (1602-1680), Leibniz idea of Geometria…

We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and also the Homeomorphism Problem for 3-manifolds and the Membership Problem for 3-manifold gr…

Optimal transport reformulates multiple quantile hedging problem.

Solves four problems related to circle families in the plane.

Solves four problems related to sphere families in 3D space.

The paper solves optimal control problems for various convex sets using convex trigonometry.

Explains eigenvalue and generalized eigenvalue problems with examples.

This paper solves the Christoffel problem in hyperbolic space and its equivalent on spheres.

In the present paper, the primal-dual problem consisting of the investment risk minimization problem and the expected return maximization problem in the mean-variance model is discussed using replica analysis. As a natural extension of the investment risk minimization problem under only a budget constraint that we anal…

Study proves only origin-centered spheres solve certain curvature problems.

MathChat uses LLM agents to solve challenging math problems through conversational problem-solving.

New algorithm solves non-convex min-max problems in signal processing.

This article reviews ranking problems and their solutions.

The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.

Paper solves Gromov-Wasserstein for point clouds efficiently.

The paper explains how microlocal analysis solves geometric inverse problems.

Proves NP and co-NP status for knot core recognition in solid torus.

A new method solves complex control problems with random coefficients.

This is a survey of some problems in geometric group theory which I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some necessary definitions and motivations, problems and some discussions of them. For ea…

We present updates to the problems on Hirzebruch's 1954 problem list focussing on open problems, and on those where substantial progress has been made in recent years. We discuss some purely topological problems, as well as geometric problems about (almost) complex structures, both algebraic and non-algebraic, about co…

27 problems identified in automating movie/TV subtitle translation.