arXiv research

Using a similar algorithm to Hatcher-Thurston's algorithm for finding a presentation of the mapping class group of a surface, Wajnryb succeeded to find a presentation for the handlebody group. This is long and complicated. In this note I simplify Wajnryb's presentation for the handlebody group of genus g = 2.

Handlebody groups reduced to 3 or 4 generators for g ≥ 5 and 3 or 4 for g ≥ 3.

S. Gervais gave a finite presentation for the mapping class group of a surface (math.GT/9811162). We show this presentation without using Wajnryb's simple presentation. We use a complex of curves defined by Harvey, in place of Hatcher and Thurston's complex.

We employ a certain labeled finite graph, called a chart, in a closed oriented surface for describing the monodromy of a(n achiral) Lefschetz fibration over the surface. Applying charts and their moves with respect to Wajnryb's presentation of mapping class groups, we first generalize a signature formula for Lefschetz …

In this paper we show that to each planar line arrangement defined over the real numbers, for which no two lines are parallel, one can write down a corresponding relation on Dehn twists that can be read off from the combinatorics and relative locations of intersections. This leads to an alternate proof of Wajnryb's gen…

Wajnryb proved that the mapping class group of an orientable surface is generated by two elements. We prove that one of these generators can be taken as a Dehn twist. We also prove that the extended mapping class group is generated by two elements, again one of which is a Dehn twist. Another result we prove is that the…

Purpose of the Conference article, intended for a wider audience, is to introduce concepts and techniques used by Bronislaw Wajnryb and the author in order to show the diffeomorphism of certain elementary algebraic surfaces, called ABC surfaces, which are not deformation equivalent. In the first part are recalled the c…

Generators found for a specific subgroup of braid groups.

Study shows monodromy kernels are large, failing to prove commensurability in specific strata.

Let $Σ_{g,p}$ be a closed oriented surface of genus $g\geq 1$ with $p$ punctures. Let $\rm Mod(Σ_{\textit{g,p}})$ be the mapping class group of $Σ_{g,p}$. Wajnryb proved in [Wa] that for $p=0, 1$ $\rm Mod({Σ_{\textit{g,p}}})$ is generated by two elements. Korkmaz proved in [Ko] that one of these generators can be taken…

We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides with the entire mapping class group of the surface. As a consequence, we construct …

We prove that there exists no a priori bound on the Euler characteristic of a closed symplectic 4-manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact 3-manifold coming from the genus of a compatible open book --- except possibly for …

New findings on generating mapping class groups using pseudo-Anosov elements.

Establishes bounds on Andrews-Curtis moves for trivial group presentations.

Study on weakly G-slim complexes and non-positive immersions for group presentations.

The abstract explores Artin presentations and their connection to 4-manifolds using triangle groups.

Investigates local indicability of groups with circle homology presentations.

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 presentaitons. Motivated by a presentation fo…

Generalized Steinberg module presentation for Gaussian and Eisenstein integers.

Finite presentation for a specific group in 3D handlebody topology.

We introduce the stable presentation length of a finitely presented group. The stable presentation length of the fundamental group of a 3-manifold can be considered as an analogue of the simplicial volume. We show that the stable presentation length have some additive properties like the simplicial volume, and the simp…

Classifies braids with positive Artin presentations and their fundamental groups.

The study shows that certain cubical presentations lead to aspherical spaces.

The paper presents methods to write presentations for Dehn quandles.

Simplified presentations for non-orientable surface mapping class groups.

Finite presentations for mapping class groups of surfaces and surfaces with points/boundaries.

Core groups are link invariants defined by arc or region presentations.

In the paper we give a survey of rather new notions and results which generalize classical ones in the theory of braids. Among such notions are various inverse monoids of partial braids. We also observe presentations different from standard Artin presentation for generalizations of braids. Namely, we consider presentat…

We give an infinite presentation for the mapping class group of a non-orientable surface with boundary components. The presentation is a generalization of the presentation given by the second author [15].

A {\em word labeled oriented graph} (WLOG) is an oriented graph $\cal G$ on vertices $X=\{ x_1,\ldots ,x_k\}$, where each oriented edge is labeled by a word in $X^{\pm1}$. WLOGs give rise to presentations which generalize Wirtinger presentations of knots. WLOG presentations, where the underlying graph is a tree are of …

New methods convert complex link presentations to simpler, recognizable forms.

We demonstrate how a 3-manifold, a Heegaard diagram, and a group presentation can each be interpreted as a pair of signed permutations in the symmetric group $S_d.$ We demonstrate the power of permutation data in programming and discuss an algorithm we have developed that takes the permutation data as input and determi…

This paper provides an infinite presentation for a subgroup of mapping class groups of non-orientable surfaces.

In this paper, we use replica analysis to determine the investment strategy that can maximize the net present value for portfolios containing multiple development projects. Replica analysis was developed in statistical mechanical informatics and econophysics to evaluate disordered systems, and here we use it to formula…

Paper proves vanishing homology groups for certain hyperbolic groups.

It is well known that surface-links in 4-space can be presented by diagrams on the plane of 4-valent spatial graphs with makers on the vertices, called marked graph diagrams. In this paper we extend the method of presenting surface-links by marked graph diagrams to presenting immersed surface-links. We also give some m…

New 3-manifold spines with unique Whitehead graphs identified.

Divide knots and links, defined by A'Campo in the singularity theory of complex curves, is a method to present knots or links by real plane curves. The present paper is a continuation of the author's previous result that every knot in the major subfamilies of Berge's lens space surgery (i.e., knots yielding a lens spac…

We present an algorithm which given a presentation of a group $G$ without 2-torsion, a solution to the word problem with respect to this presentation, and an acylindricity constant $κ$, outputs a collection of tracks in an appropriate presentation complex. We give two applications: the first is an algorithm which decid…

We give presentations, in terms of generators and relations, for the monoids of singular braids on closed surfaces. The proof of the validity of these presentations can also be applied to verify, in a new way, the presentations given by Birman for the monoids of Singular Artin braids.

Defines a cell complex for even spin mapping class group.

The paper provides infinite presentations for surface groups.

It is known that every finitely presented group is the fundamental group of the total space of a Lefschetz fibration. In this paper, we give another proof which improves the result of Korkmaz. In addition, Korkmaz defined the genus of a finitely presented group. We also evaluate upper bounds for genera of some finitely…

We define braid presentation of edge-oriented spatial graphs as a natural generalization of braid presentation of oriented links. We show that every spatial graph has a braid presentation. For an oriented link it is known that the braid index is equal to the minimal number of Seifert circles. We show that an analogy do…

The study aims to prevent unfair content presentation in recommender systems.

Alternative proof of Dynnikov's three-page index for torus links.

Consider a relatively hyperbolic group G. We prove that if G is finitely presented, so are its parabolic subgroups. Moreover, a presentation of the parabolic subgroups can be found algorithmically from a presentation of G, a solution of its word problem, and generating sets of the parabolic subgroups. We also give an a…

Presented a simple group presentation for degree four cactus group.