Related papers: Algorithmic Problems in the Braid Group
Artin's braid groups have been recently suggested as a new source for public-key cryptography. In this paper we propose the first undeniable signature schemes using the conjugacy problem and the decomposition problem in the braid groups…
In this paper we investigate the decidability and complexity of problems related to braid composition. While all known problems for a class of braids with three strands, $B_3$, have polynomial time solutions we prove that a very natural…
This paper is concerned with detecting when a closed braid and its axis are 'mutually braided' in the sense of Rudolph. It deals with closed braids which are fibred links, the simplest case being closed braids which present the unknot. The…
We consider the braid groups $\mathbf{B}_n(X)$ on finite simplicial complexes $X$, which are generalizations of those on both manifolds and graphs that have been studied already by many authors. We figure out the relationships between…
Cyclic words are equivalence classes of cyclic permutations of ordinary words. When a group is given by a rewriting relation, a rewriting system on cyclic words is induced, which is used to construct algorithms to find minimal length…
In this article, we introduce the notion of cycling operations of arbitrary order in Garside groups, which is a full generalization of the cycling and decycling operations. Theoretically, this notion together with other related concepts…
Traditional symbolic reasoning engines, while attractive for their precision and explicability, have a few major drawbacks: the use of brittle inference procedures that rely on exact matching (unification) of logical terms, an inability to…
Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid…
We construct two practical algorithms for twisted conjugacy classes of polycyclic-by-finite groups. The first algorithm determines whether two elements of a group are twisted conjugate for two given endomorphisms, under the condition that…
This paper will appear in the Santa Cruz proceedings. An overview of the braid group techniques in the theory of algebraic surfaces, from Zariski to the latest results, is presented. An outline of the Van Kampen algorithm for computing…
An element in Artin's braid group B_n is said to be periodic if some power of it lies in the center of B_n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B_n are exponential in the…
In this article, we give a numerical algorithm to compute braid groups of curves, hyperplane arrangements, and parameterized system of polynomial equations. Our main result is an algorithm that determines the cross-locus and the generators…
One of the possible generalizations of the discrete logarithm problem to arbitrary groups is the so-called conjugacy search problem (sometimes erroneously called just the conjugacy problem): given two elements a, b of a group G and the…
The cycling operation endows the super summit set $S_x$ of any element $x$ of a Garside group $G$ with the structure of a directed graph $\Gamma_x$. We establish that the subset $U_x$ of $S_x$ consisting of the circuits of $\Gamma_x$ can be…
We introduce separability properties corresponding to generalized versions of the conjugacy, twisted conjugacy, Brinkmann and Brinkmann's conjugacy problems and how they relate when finite and cyclic extensions of groups are taken. In…
The Baumslag group had been a candidate for a group with an extremely difficult word problem until Myasnikov, Ushakov, and Won succeeded to show that its word problem can be solved in polynomial time. Their result used the newly developed…
We study the conjugacy problem in cyclic extensions of free groups. It is shown that the conjugacy problem is solvable in split extensions of finitely generated free groups by virtually inner automorphisms. An algorithm for construction of…
People solve different problems and know that some of them are simple, some are complex and some insoluble. The main goal of this work is to develop a mathematical theory of algorithmic complexity for problems. This theory is aimed at…
The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to…
This paper proposes for every $n$, linear time reductions of the word and conjugacy problems on the braid groups $B_n$ to the corresponding problems on the braid monoids $B_n^+$ and moreover only using positive words representations.