Related papers: The Skein Relation for the (G_2,V)-Link Invariant
We give a simple and practical algorithm to compute the link polynomials, which are defined according to the skein relations. Our method is based on a new total order on the set of all braid representatives. As by-product a new complete…
An exposition of Vassiliev invariants is given in terms of the simplest approach to the functional integral construction of link invariants from Chern-Simons theory. This approach gives the top row evaluations of Vassiliev invariants for…
In 2018 Kashaev introduced a diagrammatic link invariant conjectured to be twice the Levine-Tristram signature. If true, the conjecture would provide a simple way of computing the Levine-Tristram signature of a link by taking the signature…
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…
Weight systems are functions on chord diagrams satisfying so-called Vassiliev's $4$-term relations. They are closely related to finite type knot invariants introduced by Vassiliev. Certain weight systems can be derived from graph…
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…
Formulas previously presented for the Casson-Walker invariant are generalized to Lescop's extension. These formulas in terms of linking numbers and surgery coefficients compute the change in Lescop's invariant under crossing changes in a…
Recently, a plethora of multivariable knot polynomials were introduced by Kashaev and one of the authors, by applying the Reshetikhin-Turaev functor to rigid $R$-matrices that come from braided Hopf algebras with automorphisms. We study the…
To a finite type knot invariant, a weight system can be associated, which is a function on chord diagrams satisfying so-called $4$-term relations. In the opposite direction, each weight system determines a finite type knot invariant. In…
We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra. The two…
The Reshetikhin - Turaev invariants arising from the quantum groups associated with the exceptional Lie algebras $G_2$, $F_4$ and $E_8$ at odd roots of unity are constructed and explicitly computed for all the lens spaces.
The theory of welded and extended welded knots is a generalization of classical knot theory. Welded (resp. extended welded) knot diagrams include virtual crossings (resp. virtual crossings and wen marks) and are equivalent under an extended…
We discuss multivariable invariants of colored links associated with the $N$-dimensional root of unity representation of the quantum group. The invariants for $N>2$ are generalizations of the multi-variable Alexander polynomial. The…
We define a variation of Khovanov homology with an explicit description in terms of the spanning trees of a link projection. We prove that this new theory is a link invariant and describe some of its properties. Finally, we provide some the…
There is a known connection between the osp(1|2n) polynomial knot invariant $J_K^n$ and the so(2n+1) knot invariant ${}_{so} J_K^n$ studied by Clark in arXiv:1509.03533 and Blumen in arXiv:0901.3232. In the rank one case, the uncolored…
This is a substantially revised version. The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev…
We generalize the braid algebra to the case of loops with intersections. We introduce the Reidemeister moves for 4 and 6-valent vertices to have a theory of rigid vertex equivalence. By considering representations of the extended braid…
We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a…
In this paper we introduce two theories of finite type invariants for framed links with fixed linking matrix. We show that these thepries are related to the theory of Vassiliev invariants of framed links. We also study the corresponding…