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Construction of (colored) knot polynomials for double-fat graphs is further generalized to the case when "fingers" and "propagators" are substituting R-matrices in arbitrary closed braids with m-strands. Original version of arXiv:1504.00371…
Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as $E^{3}$ (Euclidean 3-space), $H^{3}$ (hyperbolic 3-space) and $ E^{2,1}$ (Minkowski 3-space), using quaternion algebra theory, are…
In this paper, we construct invariants of braids, knots and links by studying dynamics of points in $\R^{2}$ and applying the Ptolemy relation $ac+bd=xy$.
We present a systematic classification of uncolored bonded knots with singularity number at most seven. Bonded knots provide a topological model for closed protein chains with intramolecular bridges, such as disulfide bonds. Following the…
We give two alternative proofs of the invariance of the Drinfeld pairing under the action of the braid group. One uses the Shapovalov form, and the other uses a characterization of the universal $R$-matrix.
We construct two families of representations of the braid group $B_n$ by considering conjugation actions on congruence subgroups of $GL_{n-1}(Z[t^{\pm 1},q^{\pm 1}])$. We show that many of these representations are faithful modulo the…
Recent years have witnessed a widespread increase of interest in network representation learning (NRL). By far most research efforts have focused on NRL for homogeneous networks like social networks where vertices are of the same type, or…
Using the theory of representations of the symmetric group, we propose an algorithm to compute the invariant ring of a permutation group. Our approach have the goal to reduce the amount of linear algebra computations and exploit a thinner…
This paper is a survey on invariants of representations of quivers and their generalizations. We present the description of generating systems for invariants and relations between generators.
Algorithm of construction of all knots, links with given number of crosses on diagram of knot, link is offered. This algorithm is based on simple proposition, that there is a representation of knot (link) as closure of braid with n threads…
A braid representation is a monoidal functor from the braid category $\mathsf{B}$, for example given by a solution to the constant Yang-Baxter equation. Given a monoidal category $\mathsf{C}$ with $ob(\mathsf{C})=\mathbb{N}$, a rank-$N$…
Any solution to the Yang-Baxter equation yields a family of representations of braid groups. Under certain conditions, identified by Turaev, the appropriately normalized trace of these representations yields a link invariant. Any…
We linearize the Artin representation of the braid group given by (right) automorphisms of a free group providing a linear faithful representation of the braid group. This result is generalized to obtain linear representations for the…
Virtual knot theory has experienced a lot of nice features that did not appear in classical knot theory, e.g., parity and picture-valued invariants. In the present paper we use virtual knot theory effects to construct new representations of…
We study combinatorial properties of virtual braid groups and we describe relations with finite type invariant theory for virtual knots and Yang-Baxter equations
Building further on work of Marin and Wagner, we give a cubic braid-type skein theory of the Links--Gould polynomial invariant of oriented links and prove that it can be used to evaluate any oriented link, adding this polynomial to the list…
The paper is devoted to invariant theory problems. In particular, to the problem of finding generators of invariant fields in an explicit form. The set of generators is given for invariant field of unitriangular group of adjoint…
We study invariant operators in general tensor models. We show that representation theory provides an efficient framework to count and classify invariants in tensor models. In continuation and completion of our earlier work, we present two…
The notion of a braided chord diagram is introduced and studied. An equivalence relation is given which identifies all braidings of a fixed chord diagram. It is shown that finite-type invariants are stratified by braid index for knots which…
In 1994, Long and Moody introduced a method to construct a new representation of the braid group from the representation of the braid group or the semidirect product of the braid group and the free group. In this paper, we show that its…