Related papers: On the Kauffman-Jones polynomial for virtual singu…
We extend the state models for Jones and Alexander polynomials of classical links to state models of 2-variable polynomials in the case of singular links. Moreover, we extend both of them to polynomials with d+1 variables for long singular…
A polynomial invariant of virtual links, arising from an invariant of links in thickened surfaces introduced by Jaeger, Kauffman, and Saleur, is defined and its properties are investigated. Examples are given that the invariant can detect…
We give a congruence relating a one variable specialization of the two variable Kauffman polynomial of any periodic link to that of its mirror image. Consequently, we obtain a new and simple criterion for periodicity of links.
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…
In the present paper, we construct the Khovanov homology theory for virtual links. Besides the direct approach with Z_{2} coefficients we also describe the Khovanov homology for framed links and the Khovanov homology using ``double cover''.…
We construct a 2-variable link polynomial, called $W_L$, for classical links by considering simultaneously the Kauffman state models for the Alexander and for the Jones polynomials. We conjecture that this polynomial is the product of two…
We construct new invariant polynomial for long virtual knots. It is a generalization of Alexander polynomial. We designate it by $\zeta$ meaning an analogy with $\zeta$-polynomial for virtual links. A degree of $\zeta$-polynomial estimates…
Two categorifications are given for the arrow polynomial, an extension of the Kauffman bracket polynomial for virtual knots. The arrow polynomial extends the bracket polynomial to infinitely many variables, each variable corresponding to an…
Recently, for a limited class for bipartite links, the complicated Khovanov-Rozansky matrix factorization technique was reduced to an analogue of elementary Kauffman-Khovanov cycle calculus for an arbitrary $N$. In this note, we demonstrate…
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 weak chord index $Ind'$ is constructed for self crossing points of virtual links. Then a new writhe polynomial $W$ of virtual links is defined by using $Ind'$. $W$ is a generalization of writhe polynomial defined in [6]. Based on $W$,…
This paper contains general formulae for the reduced relative Tutte, Kauffman bracket and Jones polynomials of families of virtual knots and links given in Conway notation and discussion of a counterexample to the Z-move conjecture of Fenn,…
The theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from sl(2) to sl(N) at arbitrary N -- but…
We extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex ($h=t=0$), the variant of Lee ($h=0,t=1$) and other…
We extend the cobordism based categorification of the virtual Jones polynomial to virtual tangles. This extension is combinatorial and has semi-local properties. We use the semi-local property to prove an applications, i.e. we give a…
This paper gives a polynomial invariant for flat virtual links. In the case of one component, the polynomial specializes to Turaev's virtual string polynomial. We show that Turaev's polynomial has the property that it is non-zero precisely…
The virtual skein relation for the Jones polynomial of the virtual link diagram was introduced by N. Kamada, S. Nakabo, and S. Satoh. H. A. Dye, L. H. Kauffman, and Y. Miyazawa introduced multivariable polynomial, an invariant of virtual…
Flat virtual links are some variant of links, and semiquandles are counterparts of quandles or biquandles, which axiomize the Reidemeister-like moves. In this paper, we give some example of semiquandle and introduce an invariant for flat…
We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also…
We give an upper bound on the z-degree of the Kauffman polynomial of a link, using bridges of length greater than one which are separated in some tangle decomposition of a link diagram. We construct some examples by wiring together rational…