Narrow Results

Omori and the author [R. Kobayashi and G. Omori, An infinite presentation for the mapping class group of a non-orientable surface with boundary, Osaka J. Math. 59(2) (2022) 269–314] have given an infinite presentation for the mapping class group of a compact non-orientable surface. In this paper, we give more simple infinite presentations for this group.

We proceed with the investigation of a method of quantization of the observable sector of closed bosonic strings. For the presentation of the quantum algebra of observables the construction cycle concerning elements of order ℏ⁶ has been carried out. We have computed the quantum corrections to the only generating relation of order ℏ⁶. This relation is of spin-parity J^P = 0⁺. We found that the quantum corrections to this relation break the semidirect splitting of the classical algebra into an Abelian, infinitely generated subalgebra and a non-Abelian, finitely generated subalgebra . We have established that there are no ("truly independent") generating relations of order ℏ⁷.

We study the way in which the abstract structure of a small overlap monoid is reflected in, and may be algorithmically deduced from, a small overlap presentation. We show that every C(2) monoid admits an essentially canonical C(2) presentation; by counting canonical presentations we obtain asymptotic estimates for the number of non-isomorphic monoids admitting a-generator, k-relation presentations of a given length. We demonstrate an algorithm to transform an arbitrary presentation for a C(m) monoid (m at least 2) into a canonical C(m) presentation, and a solution to the isomorphism problem for C(2) presentations. We also find a simple combinatorial condition on a C(4) presentation which is necessary and sufficient for the monoid presented to be left cancellative. We apply this to obtain algorithms to decide if a given C(4) monoid is left cancellative, right cancellative or cancellative, and to show that cancellativity properties are asymptotically visible in the sense of generic-case complexity.

Let $G$ be a group and let $x\in G$ be a left $3$-Engel element of order dividing $60$. Suppose furthermore that ${〈x〉}^G$ has no elements of order $8$, $9$ and $25$. We show that $x$ is then contained in the locally nilpotent radical of $G$. In particular, all the left $3$-Engel elements of a group of exponent $60$ are contained in the locally nilpotent radical.

In this paper, we produce the product rules of nonassociative Moufang loops of order 81 by using an analytical approach. We then explore all possible presentations on a suitable set of generators, thereby obtaining a total of five nonisomorphic cases. The result is in agreement with the classification by Nagy and Vojtěchovský using the GAP package LOOPS.

Metric conditions "C_α" and "" are defined for finite group presentations. If the fundamental group of a closed aspherical 3-manifold has some presentation which satisfies C₂ or , then its universal cover is simply connected at infinity. These ideas are derived from work by A. Casson and V. Poénaru.

It is proved that the Virasoro algebra can be constructed starting with only two generators L₃, L₋₂ subjected to 5 relations without neither introducing a central element c (charge) nor assuming any symmetry relation. Our proof is supported by substantial symbolic computations in REDUCE.

In this paper we give new presentations of the braid groups and the pure braid groups of a closed surface. We also give an algorithm to solve the word problem in these groups, using the given presentations.

We show that any tunnel number one knot group has a two generator one relator presentation in which the relator is a palindrome in the generators. We use this fact to compute the character variety for this knot groups and we show that it is an affine algebraic set .

Let f: K² → M³ be an embedding of a compact, connected 2-complex into a compact, connected, orientable 3-manifold. We are interested to know, to which extent K² determines (up to homeomorphism) its regular neighborhood N(f(K²)) ⊆ M³. We exhibit a list of four obstructions to uniqueness, and prove that in the absence of these, the regular neighborhoods are uniquely determined: Let f,K²,M³ be as above and assume M³ is prime and not a Poincaré-counterexample. If N(f(K²)) does not contain essential annuli and has connected boundary, then N is determined by K².

The inverse braid monoid consists of n-string braids from which some subset of the strings has been deleted. Such partial braids may be composed by concatenation followed by the deletion of any incomplete strings, and this operation gives the structure of an inverse monoid. We derive presentations of by first using a groupoid presentation, and then deriving equivalent inverse monoid presentations. In particular, we offer a new derivation of the presentation for that has been given by Easdown and Lavers.

We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged "convexly", which is why we describe them as geometric presentations. Motivated by a presentation for the full braid group that we call the "rotation presentation", we introduce presentations for the pure braid group that we call the "twist presentation" and the "swing presentation". From the point of view of mapping class groups, the swing presentation can be interpreted as stating that the pure braid group is generated by a finite number of Dehn twists and that the only relations needed are the disjointness relation and the lantern relation.

Braid groups and mapping class groups have many features in common. Similarly to the notion of inverse braid monoid, inverse mapping class monoid is defined. It concerns surfaces with punctures, but among given n punctures, several can be omitted. This corresponds to braids where the number of strings is not fixed. In the paper we give the analogue of the Dehn–Nilsen–Baer theorem, propose a presentation of the inverse mapping class monoid for a punctured sphere and study the word problem. This shows that certain properties and objects based on mapping class groups may be extended to the inverse mapping class monoids. We also give analogues of Artin presentation with two generators.

In this paper, we introduce ${\mathbb{Z}}_2$-braids and, more generally, $G$-braids for an arbitrary group $G$. They form a natural group-theoretic counterpart of $G$-knots, see [V. O. Manturov; Reidemeister moves and groups, preprint (2014), arXiv:1412.8691]. The underlying idea used in the construction of these objects — decoration of crossings with some additional information — generalizes an important notion of parity introduced by the second author (see [V. O. Manturov, Parity in knot theory, Sb. Math.201(5) (2010) 693–733]) to different combinatorically geometric theories, such as knot theory, braid theory and others. These objects act as natural enhancements of classical (Artin) braid groups. The notion of dotted braid group is introduced: classical (Artin) braid groups live inside dotted braid groups as those elements having presentation with no dots on the strands. The paper is concluded by a list of unsolved problems.

Scaphoid injury and subsequent non-union is a well documented and researched subject. This article gives an overview of the epidemiology and results of the patients we have treated for scaphoid non-union at a University Hospital. 283 scaphoid non-unions in 268 patients (83% men) were operated upon, 230 as a primary and 47 as a secondary. The median age at time of surgery was 27 years. One-third of the non-unions were located in the proximal part of the scaphoid and the remaining two-thirds in the middle part. Of the 146 patients (55%) who contacted a doctor at the time of injury, 53 fractures where diagnosed (20%). Fourteen (5%) were operated primarily while 39 (15%) (seven dislocated) were immobilized in plaster casts. Thirty-two (12%) were under the age of 16 at the time of injury. The average time from the injury to the initial non-union surgery was 1.5 years with 2.8 years to the second procedure. The risk of osteoarthritis increased time from injury to surgery (both primary and secondary procedures). The greatest potential for the reduction of scaphoid non-union is an increased awareness amongst younger men. There is also potential for improved accuracy in the diagnosis of scaphoid fractures (better clinical tests, the use of radiographs, CTs and MRIs) in order to identify the fracture and evaluate dislocation at the initial injury. Early diagnosis and treatment of fractures and non-unions will reduce the development of degenerative wrist changes.

We obtain the explicit generators of the annihilator ideal for the tensor product of any finite dimensional simple module over quantum group , by using the weight property of ideals in when q is not a root of unity. As an application, we give a presentation of quantum generalized Schur algebra.

The symplectic blob algebra b_n(n ∈ ℕ) is a finite-dimensional algebra defined by a multiplication rule on a basis of certain diagrams. The rank r(n) of b_n is not known in general, but r(n)/n grows unboundedly with n. For each b_n we define an algebra by presentation, such that the number of generators and relations grows linearly with n. We prove that these algebras are isomorphic.

In this work, we prove that if a triangular algebra $A$ admits a strongly simply connected universal Galois covering for a given presentation, then the fundamental group associated to this presentation is free.

Let M be a monoid and let θ be an endomorphism of M. We prove that if the Bruck–Reilly extension BR(M, θ) is finitely presented, then the Bruck–Reilly extension BR(M, θ^m) is also finitely presented for all m ⩾ 1.

A fundamental theory on the behavior and properties of trivializers of 2-complexes is set up which can be applied to deal with homotopy problems of group presentations and monoid presentations.