Related papers: A new two-variable generalization of the Jones pol…
The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type…
In this note we define a polynomial invariant for colored links by a skein relation. It specializes to the Jones polynomial for classical links.
We give characterizations of the skein polynomial for links (as well as Jones and Alexander-Conway polynomials derivable from it), avoiding the usual "smoothing of a crossing" move. As by-products we have characterizations of these…
Given any oriented link diagram, one can construct knot invariants using skein relations. Usually such a skein relation contains three or four terms. In this paper, the author introduces several new ways to smooth a crossings, and uses a…
Given any unoriented link diagram, a group of new knot invariants are constructed. Each of them satisfies a generalized 4 term skein relation. The coefficients of each invariant is from a commutative ring. Homomorphisms and representations…
The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman…
It is a well known result from Thistlethwaite that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the…
Measuring the entanglement complexity of collections of open curves in 3-space has been an intractable, yet pressing mathematical problem, relevant to a plethora of physical systems, such as in polymers and biopolymers. In this manuscript,…
Given any diagram of a link, we define on the cube of Kauffman's states a "2-complex" whose homology is an invariant of the associated framed links, and such that the graded Euler characteristic reproduces the unnormalized Kauffman bracket.…
This paper continues the study of periodic links started in \cite{Politarczyk2}. It contains a study of the equivariant analogues of the Jones polynomial, which can be obtained from the equivariant Khovanov homology. In this paper we…
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…
We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a…
This paper defines versions of the Jones polynomial and Khovanov homology by using several maps from the set of Gauss diagrams to its variant. Through calculation of some examples, this paper also shows that these versions behave…
We study relationships between the colored Jones polynomial and the A-polynomial of a knot. We establish for a large class of 2-bridge knots the AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the A-polynomial.…
The Jones polynomial $V_{L}(t)$ for an oriented link $L$ is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer $n\ge 3$, we show that: (1) the difference of Jones polynomials for two oriented links which…
The present paper is an introduction to a combinatorial theory arising as a natural generalisation of classical and virtual knot theory. There is a way to encode links by a class of `realisable' graphs. When passing to generic graphs with…
New invariants of links are constructed using the skein invariant polynomial of colored links defined by the author in [1]. These invariants are stronger than the homflypt polynomial.
We generalize Kauffman's famous formula defining the Jones polynomial of an oriented link in 3-space from his bracket and the writhe of an oriented diagram. Our generalization is an epimorphism between skein modules of tangles in compact…
We explain how to compute the Jones polynomial of a link from one of its grid diagrams and we observe a connection between Bigelow's homological definition of the Jones polynomial and Kauffman's definition of the Jones polynomial.…
This paper is a self-contained development of an invariant of graphs embedded in three-dimensional Euclidean space using the Jones polynomial and skein theory. Some examples of the invariant are computed. An unlinked embedded graph is one…