R-matrix knot invariants and triangulations

We have new solutions to the Yang-Baxter equation, from which we have constructed new link invariants containing more than two arbitrary parameters. This may be regarded as a generalization of the Jones' polynomial. We have also found another simpler invariant which discriminates only the linking structure of knots with each other, but not details of individual knot.

We define quantum matrix groups GL(3) by their coaction on appropriate quantum planes and the requirement that the Poincare series coincides with the classical one. It is shown that this implies the existence of a Yang-Baxter operator. Exploiting stronger equations arising at degree four of the algebra, we classify all quantum matrix groups GL(3). We find 26 classes of solutions, two of which do not admit a normal ordering. The corresponding R-matrices are given.

We describe the construction of trigonometric R-matrices corresponding to the (multiplicity-free) tensor product of any two irreducible representations of a quantum algebra $U_q(\G)$. Our method is a generalization of the tensor product graph method to the case of two different representations. It yields the decomposition of the R-matrix into projection operators. Many new examples of trigonometric R-matrices (solutions to the spectral parameter dependent Yang-Baxter equation...

Boundary solutions to the quantum Yang-Baxter (qYB) equation are defined to be those in the boundary of (but not in) the variety of solutions to the ``modified'' qYB equation, the latter being analogous to the modified classical Yang-Baxter (cYB) equation. We construct, for a large class of solutions $r$ to the modified cYB equation, explicit ``boundary quantizations'', i.e., boundary solutions to the qYB equation of the form $I+tr+ t^2r_{2} + ...$. In the last section we lis...

We construct knot invariants on the basis of ascribing Euclidean geometric values to a triangulation of sphere S^3 where the knot lies. The main new feature of this construction compared to the author's earlier papers on manifold invariants is that now nonzero "deficit angles" (in the terminology of Regge calculus) can also be handled. Moreover, the knot goes exactly along those edges of triangulations that have nonzero deficit angles.

It has been a long-standing problem how to relate Chern-Simons theory to the quantum groups. In this paper we recover the classical $r$-matrix directly from a 3-dimensional Chern-Simons theory with boundary conditions, thus creating a direct link to the quantum groups. It is known that the Jones polynomials can be constructed using an $R$-matrix. We show how these constructions can be seen to arise directly from 3-dimensional Chern-Simons theory.

One of the principal obstacles on the way to quantum computers is the lack of distinguished basis in the space of unitary evolutions and thus the lack of the commonly accepted set of basic operations (universal gates). A natural choice, however, is at hand: it is provided by the quantum R-matrices, the entangling deformations of non-entangling (classical) permutations, distinguished from the points of view of group theory, integrable systems and modern theory of non-perturbat...

A general functional definition of the infinite dimensional quantum R-matrix satisfying the Yang-Baxter equation is given. A procedure for extracting a finite dimensional R-matrix from the general definition is demonstrated for the particular cases of the group O(2) and of the group of translations.

In the classification of solutions of the Yang--Baxter equation, there are solutions that are not deformations of the trivial solution (essentially the identity). We consider the algebras defined by these solutions, and the corresponding dual algebras. We then study the representations of the latter. We are also interested in the Baxterisation of these $R$-matrices and in the corresponding quantum planes.

A general functional definition of the infinite dimensional quantum $R$-matrix satisfying the Yang-Baxter equation is given. A procedure for the extracting a finite dimensional $R$-matrix from the general definition is demonstrated in a particular case when the group $SU(2)$ takes place.