Related papers: A Sequence of Degree One Vassiliev Invariants for …
As it is well-known, all Vassiliev invariants of degree one of a knot $K\subset R^3$ are trivial. There are nontrivial Vassiliev invariants of degree one, when the ambient space is not $R^3$. Recently, T. Fiedler introduced such invariants…
We extend to the long virtual knot case the constructions first presented by A. Henrich and later generalized by the author to the framed virtual knot case. These consist of three Vassiliev invariants of order one, including a universal…
We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in arXiv:1509.00578. In particular, following the theory of quantum invariants we work with 'rotational' virtual…
It is an open question whether there are Vassiliev invariants that can distinguish an oriented knot from its inverse, i.e., the knot with the opposite orientation. In this article, an example is given for a first order Vassiliev invariant…
Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas have been unknown explicitly, where only one known formula was…
We introduce a theory of virtual Legendrian knots. A virtual Legendrian knot is a cooriented wavefront on an oriented surface up to Legendrian isotopy of its lift to the unit cotangent bundle and stabilization and destablization of the…
We define and study Vassiliev invariants for (long) Morse knots. It is shown that there are Vassiliev invariants which can distinguish some topologically equivalent Morse knots. In particular, there is an invariant of order 3 for Morse…
It is well-known that self-linking is the only Z valued Vassiliev invariant of framed knots in S^3. However for most 3-manifolds, in particular for the total spaces of S^1-bundles over an orientable surface F not S^2, the space of Z-valued…
This paper introduces two virtual knot theory ``analogues'' of a well-known family of invariants for knots in thickened surfaces: the Grishanov-Vassiliev finite-type invariants of order two. The first, called the three loop isotopy…
A. Henrich proved the existence of the universal finite-type invariant of order one for virtual knots. We extend the construction and the methods of her paper to framed virtual knots. To do so, we introduce the notions of virtual strings…
We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite…
Oleg Viro introduced an invariant of rigid isotopy for real algebraic knots in $RP^3$ which can be viewed as a first order Vassiliev invariant. In this paper we look at real algebraic knots of degree $d$ with the maximal possible value of…
We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are…
Vassiliev's knot invariants can be computed in different ways but many of them as Kontsevich integral are very difficult. We consider more visual diagram formulas of the type Polyak-Viro and give new diagram formula for the two basic…
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…
We introduce two polynomial invariants $V_1(K;t)$ and $V_2(K;t)$ of a long virtual knot $K$, which generalize the degree-two finite type invariants $v_{2,1}$ and $v_{2,2}$ of Goussarov, Polyak, and Viro. We establish their fundamental…
A singular knot is an immersed circle in $\mathbb R^{3}$ with finitely many transverse double points. The study of singular knots was initially motivated by the study of Vassiliev invariants. Namely, singular knots give rise to a decreasing…
In this paper, we introduce the 0-smoothing invariant $\mathcal{F}$ of virtual knotoids constructed from local modification at classical crossings, which take values in a free $\mathbb Z$-module generated by non-oriented flat virtual…
We refine the Polyak-Viro Gauss diagram formula for the Vassiliev invariant of order two in a very simple way for the 2-cable of a framed long knot. Surprisingly, the resulting isotopy invariant of framed knots can detect already the…
In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of…