Related papers: A user's guide to basic knot and link theory
In this paper we construct new invariants of knotoids including the odd writhe, the parity bracket polynomial, the affine index polynomial and the arrow polynomial, and give an introduction to the theory of virtual knotoids. The invariants…
The alternating knots, links and twists projected on the $S_2$ sphere were identified with the phase space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossings, the edges correspond…
The motivation for this work is to construct a map from classical knots to virtual ones. What we get in the paper is a series of maps from knots in the full torus (thickened torus) to flat-virtual knots. We give definition of flat-virtual…
We give a rational surgery formula for the Casson-Walker invariant of a 2-component link in $S^{3}$ which is a generalization of Matveev-Polyak's formula. As application, we give more examples of non-hyperbolic L-space $M$ such that knots…
The Alexander theorem (1923) and the Markov theorem (1936) are two classical results in knot theory that show respectively that every link is the closure of a braid and that braids that have the same closure are related by a finite number…
Gauss diagram formulas are extensively used to study Vassiliev link invariants. Now we apply this approach to invariants of 3-manifolds, considering manifolds given by surgery on framed links in the 3-sphere. We study the lowest degree case…
Twisted Alexander invariants of knots are well-defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the…
This paper contains linear systems of equations which can distinguish knots without knot invariants. Let $M_n$ be the topological moduli space of all n-component string links and such that a fixed projection into the plane is an immersion.…
It is well known how the linking number and framing can be extracted from the degree 1 part of the (framed) Kontsevich integral. This note gives a general formula expressing any product of powers of these two invariants as combination of…
We extend the results of our previous paper from knots to links by using a formula for the Jones polynomial of a link derived recently by N. Reshetikhin. We illustrate this formula by an example of a torus link. A relation between the…
We publish a table of primitive finite-type invariants of order less than or equal to six, for knots of ten or fewer crossings. We note certain mod-2 congruences, one of which leads to a chirality criterion in the Alexander polynomial. We…
The Links--Gould invariant $\mathrm{LG}(L ; t_0, t_1)$ of a link $L$ is a two-variable quantum generalization of the Alexander--Conway polynomial $\Delta_L(t)$ and has been shown to share some of its most geometric features in several…
The theory of link-homotopy, introduced by Milnor, is an important part of the knot theory, with Milnor's mu-bar-invariants being the basic set of link-homotopy invariants. Skein relations for knot and link invariants played a crucial role…
We study the Vassiliev knot invariant v_2 of degree 2. We present it via the degrees of maps of various configuration spaces related to a knot to products of spheres. This gives rise to numerous geometrical and combinatorial formulas for…
Given a finite dimensional representation of a semisimple Lie algebra there are two ways of constructing link invariants: 1) quantum group invariants using the R-matrix, 2) the Kontsevich universal link invariant followed by the Lie algebra…
The purpose of the paper is twofold. First, we give a short proof using the Kontsevich integral for the fact that the restriction of an invariant of degree 2n to (n+1)-component Brunnian links can be expressed as a quadratic form on the…
We formulate the holographic principle for knots and links. For the "space" of all knots and links, torus knots T(2m+1,2) and torus links L(2m,2) play the role of the "boundary" of this space. Using the holographic principle, we find the…
In this work we demonstrate that the q-numbers and their two-parameter generalization, the q,p-numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely…
We introduce two new families of polynomial invariants of oriented classical and virtual knots and links defined as decategorfications of the quandle coloring quiver. We provide examples to illustrate the computation of the invariants, show…
In this paper, we formulate a construction of ideal coset invariants for surface-links in $4$-space using invariants for knots and links in $3$-space. We apply the construction to the Kauffman bracket polynomial invariant and obtain an…