相关论文: Coloring Torus Knots by Conjugation Quandles

This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let \pi_K be…

Let $G$ be the fundamental group of the complement of the torus knot of type $(m,n)$. This has a presentation $G=<x,y|x^m=y^n>$. We find the geometric description of the character variety $X(G)$ of characters of representations of $G$ into…

We study the unwheeled rational Kontsevich integral of torus knots. We give a precise formula for these invariants up to loop degree 3 and show that they appear as colorings of simple diagrams. We show that they behave under cyclic branched…

We conjecture the existence of four independent gradings in the colored HOMFLY homology. We describe these gradings explicitly for the rectangular colored homology of torus knots and make qualitative predictions of various interesting…

We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of Colorings beyond Fox: The other linear Alexander quandles (Linear Algebra and its Applications, Vol. 548,…

In this article we take up the calculation of the minimum number of colors needed to produce a non-trivial coloring of a knot. This is a knot invariant and we use the torus knots of type (2, n) as our case study. We calculate the minima in…

A quandle is an algebraic system whose axioms are motivated by Reidemeister moves in knot theory. A typical example is a conjugation quandle arising from a group. A quandle is said to be admissible if it is isomorphic to a conjugation…

The knot group is the fundamental group of a knot or link complement. A necessary and sufficient conditions for a group to be realized as the knot group of some link was provided. This result was shown using the closed braid method.…

We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles whose components are labelled by irreducible representations of U_q(sl(2)). We show that the corresponding colored invariants of tangles can be…

Every torus knot can be represented as a Fourier-(1,1,2) knot which is the simplest possible Fourier representation for such a knot. This answers a question of Kauffman and confirms the conjecture made by Boocher, Daigle, Hoste and Zheng.…

Quandles are certain algebraic structures showing up in different mathematical contexts. A group $G$ with the conjugation operation forms a quandle, $\operatorname{Conj}(G)$. In the opposite direction, one can construct a group…

We enhance the quandle coloring quiver invariant of oriented knots and links with quandle modules. This results in a two-variable polynomial invariant with specializes to the previous quandle module polynomial invariant as well as to the…

Lower bounds of betti numbers for homology groups of racks and quandles will be given using the quotient homomorphism to the orbit quandles. Exact sequences relating various types of homology groups are analyzed. Geometric methods of…

In this paper we study the knot Floer homology of a subfamily of twisted $(p, q)$ torus knots where $q \equiv\pm1$ (mod $p$). Specifically, we classify the knots in this subfamily that admit L-space surgeries. To do calculations, we use the…

We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most…

We compute the triply graded Khovanov-Rozansky homology of a family of links, including positive torus links and $\operatorname{Sym}^l$-colored torus knots.

We study the invariant of knots in lens spaces defined from quantum Chern-Simons theory. By means of the knot operator formalism, we derive a generalization of the Rosso-Jones formula for torus knots in L(p,1). In the second part of the…

For p=3 and for p=5 we prove that there are exactly p equivalence classes of p-coloured knots modulo (+/-1)--framed surgeries along unknots in the kernel of a p-colouring. These equivalence classes are represented by connect-sums of n…

Given knots K and J, one can ask whether a single smoothing of a crossing in a diagram for K can convert it into a diagram for J. As an interesting example, Zekovic discovered that the torus knot T(2,5) can be converted into T(2,-5) with a…

Knot concordance plays a crucial role in the low dimensional topology. We propose a very elementary techniques which allows one to construct a lot of sliceness obstructions for knots in the full torus. Our approach deals with group…