Related papers: Symbolic computation with finite quandles
Geometric representations of cycles in quandle homology theory are given in terms of colored knot diagrams. Abstract knot diagrams are generalized to diagrams with exceptional points which, when colored, correspond to degenerate cycles.…
Enveloping $C^*$-algebras for some finitely generated $*$-algebras are considered. It is shown that all of the considered algebras are identically defined by their dual spaces. The description in terms of matrix-functions is given. Keywords…
A quandle is an algebraic structure whose axioms correspond to the Reidemeister moves of knot theory. S. Kamada introduced the notion of a quandle with a good involution, which is later called a symmetric quandle. We are interested in the…
In our previous paper math/0502157 we classified a large class of finite-dimensional pointed Hopf algebras up to isomorphism. However the following problem was left open for Hopf algebras of of type $A,D$ or $E_6$, that is whose Cartan…
The idea of using unfolding as a way of computing a program semantics has been applied successfully to logic programs and has shown itself a powerful tool that provides concrete, implementable results, as its outcome is actually source…
We define the notion of the orbit group of a quandle via its connectivity and compute the orbit groups for some basic quandles. We also show that the orbit group counts the number of orbits of certain quandles.
We study the problem of finding a subgroup of a given order in a finite group, where the group is represented by its Cayley table. We analyze the complexity of the problem in the special case of abelian groups and present an optimal…
A quandle will be called quasi-affine, if it embeds into an affine quandle. Our main result is a characterization of quasi-affine quandles, by group-theoretic properties of their displacement group, by a universal algebraic condition coming…
In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences,…
The rank of a finite algebraic structure with a single binary operation is the minimum number of elements needed to express every other element under the closure of the operation. In the case of groups, the previous best algorithm for…
Counting the number of perfect matchings in bipartite graphs, or equivalently computing the permanent of 0-1 matrices, is an important combinatorial problem that has been extensively studied by theoreticians and practitioners alike. The…
Originally, tangles were invented as an abstract tool in mathematical graph theory to prove the famous graph minor theorem. In this paper, we showcase the practical potential of tangles in machine learning applications. Given a collection…
In this paper, we introduce the notion of circular orderability for quandles. We show that the set all right (respectively left) circular orderings of a quandle is a compact topological space. We also show that the space of right…
In Part I of this series we described three algorithms that construct canonical tree-decompositions of graphs which distinguish all their k-blocks and tangles of order k. We now establish bounds on the number of parts in these…
We relate the computational complexity of finite strings to universal representations of their underlying symmetries. First, Boolean functions are classified using the universal covering topologies of the circuits which enumerate them. A…
Many standard structural quantities, such as order parameters and correlation functions, exist for common condensed matter systems, such as spherical and rod-like particles. However, these structural quantities are often insufficient for…
We introduce the notion of a hierarchical quandle, which is a generalisation of diquandles and multi-quandles. Using hierarchical quandle colourings, we construct a cocycle invariants for links coloured by quandles.
The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the…
This paper talk about the complexity of computation by Turing Machine. I take attention to the relation of symmetry and order structure of the data, and I think about the limitation of computation time. First, I make general problem named…
Ordered binary decision diagrams (OBDDs) are a fundamental data structure for the manipulation of Boolean functions, with strong applications to finite-state symbolic model checking. OBDDs allow for efficient algorithms using top-down…