arXiv research

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

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…

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…

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…

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

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

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

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

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

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 $\fsl_3$, …

The paper calculates colored Jones polynomials for specific link configurations.

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…

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 …

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…

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…

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…

Researchers compute and predict knot volumes using colored Jones polynomials.

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 …

We demonstrate that three maximal subgroups of infinite index in the rectangular subgroup \( K_{(2,2)} \) of the Thompson group \( F \), each containing Jones's \( 3 \)-colorable subgroup \( \mathcal{F} \), can be characterized as stabilizer subgroups. Additionally, we show that the \( \vec{F} \)-index, an elementary k…

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…

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

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

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

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…

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…

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

Hard instances, which require a long time for a specific algorithm to solve, help (1) analyze the algorithm for accelerating it and (2) build a good benchmark for evaluating the performance of algorithms. There exist several efforts for automatic generation of hard instances. For example, evolutionary algorithms have b…

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

The paper calculates actions of string link operations for 4- and 5-component links.

Study proves chainmail links are L-space links.

New findings on T-links derived from torus links.

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

Paper finds linking numbers for Montesinos links using a simple algorithm.

We define and prove properties of link lattice complexes for plumbed links.

Study of Lorenz links and T-links, showing equivalence and unique presentations.

Paper proves Reshetikhin-Turaev link invariants appear in higher order terms of re-normalized link invariants for plumbed links.

Extends positive and almost positive links to successively almost positive ones.

Study of knots and links in 2-complexes, defining linking numbers and polynomials.

A virtual link is a generalization of a classical link that is defined as an equivalence class of certain diagrams, called virtual link diagrams. It is further generalized to a twisted link. Twisted links are in one-to-one correspondence with stable equivalence classes of links in oriented thickenings of (possibly non-…

We generalized the periodic links to \emph{transitive} links in a $3$-manifold $M$. We find a complete classification theorem of transitive links in a $3$-dimensional sphere $\mathbb{R}^3$. We study these links from several different aspects including polynomial invariants using the relation between link polynomials of…

Characterizes a subset of links using quasipositive and homogeneous properties.

Study links' flat-virtual diagrams to create link invariants.

New link invariants from diagram colorings match link widths.

Innovates a three-component link homotopy invariant.

Enhanced Alexander module detects linking numbers in links.

The theory of signature invariants of links in rational homology spheres is applied to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, an explicit formula is derived to compute signature invariants of their covering links. Using the formula, we produce fused bou…

New methods show hyperbolicity of Brunnian links.

A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative. Normal virtual links have some properties similar to classical links.In this paper, we introduce a method of converting a virtual link d…