This paper finds upper bounds for lattice stick numbers of rational links with specific stick configurations.
problem Finding upper bounds for the lattice stick number of rational links with exactly 4 z-sticks.
method Using 2-circuit presentations, the paper constructs lattice stick numbers with exactly 4 z-sticks and derives upper bounds.
result Upper bounds for the lattice stick number of rational links with exactly 4 z-sticks are derived.
Establishes bounds on Andrews-Curtis moves for trivial group presentations.
problem Understanding presentations of the trivial group and Andrews-Curtis moves.
method Explicit upper bounds on stable Andrews-Curtis moves for thickenable presentations.
result Thickenable presentations of the trivial group satisfy the Andrews-Curtis conjecture.
Study on weakly G-slim complexes and non-positive immersions for group presentations.
problem Conditions for non-positive immersions in group presentations.
method Investigation of weakly G-slim complexes and their relationship to left-orderable groups.
result Conditions on generalized Wirtinger presentations guaranteeing non-positive immersions for their associated 2-complexes.
The abstract explores Artin presentations and their connection to 4-manifolds using triangle groups.
problem Characterizing and classifying 4-manifolds using Artin presentations and triangle groups.
method Utilizing triangle groups to find Artin presentations that present the trivial group and determining 4-manifolds with specific properties.
result Identified all Artin presentations on two generators that present the trivial group and all smooth, closed, simply-connected 4-manifolds with specific properties.
Investigates local indicability of groups with circle homology presentations.
problem Conditions for local indicability in groups with circle homology presentations.
method Generalizes results for two-relator presentations to circle homology presentations.
result Extends results on local indicability to LOT groups and non-cycle-free Adian 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.
problem Presenting Steinberg modules for specific number rings.
method Generalization of Bykovskii's presentation to Gaussian and Eisenstein integers.
result Generalization does not yield a presentation for all Euclidean number rings.
Finite presentation for a specific group in 3D handlebody topology.
problem Finding a finite presentation for a specific group in 3D handlebody topology.
method Constructed a finite presentation for the liftable Hilden group to derive the presentation for the balanced superelliptic handlebody group.
result A finite presentation was given for the balanced superelliptic handlebody group.
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.
problem Classifying braids with positive Artin presentations and understanding their fundamental groups.
method Analyzing framed, closed pure n-braids in the 3-sphere to determine if they represent positive Artin presentations.
result Closed, pure n-braids B' in the 3-sphere that represent positive Artin presentations are strongly invertible, and some 3-manifolds do not admit such presentations.
The study shows that certain cubical presentations lead to aspherical spaces.
problem Understanding the asphericity of cubical presentations in 2D.
method Analyzing the second homotopy group of coned-off spaces associated with cubical presentations.
result The coned-off space is aspherical under specific conditions.
The paper presents methods to write presentations for Dehn quandles.
problem Writing explicit presentations for Dehn quandles.
method Two approaches: for Garside and Gaussian groups, and for general Dehn quandles with known centralisers.
result Examples of Dehn quandles including spherical Artin groups, surface groups, and mapping class groups.
Paper uses replica analysis to optimize net present value in investment portfolios.
problem Maximizing net present value in portfolios of multiple development projects.
method Replica analysis applied to optimization problem with budget and investment constraints.
result Replica analysis yields higher net present value than conventional methods.
Simplified presentations for non-orientable surface mapping class groups.
problem Presenting the mapping class group of non-orientable surfaces.
method Provided simpler infinite presentations.
result Simplified presentations for the group.
Finite presentations for mapping class groups of surfaces and surfaces with points/boundaries.
problem Finding finite presentations for balanced superelliptic mapping class groups.
method Construct finite presentations for corresponding liftable mapping class groups in a different generating set.
result Finite presentations for balanced superelliptic mapping class groups of various surfaces.
Goeritz and Seifert matrices derived from Dehn presentations.
problem Computing Goeritz and Seifert matrices for links.
method Using Fox's free differential calculus on modified Dehn presentations.
result Goeritz and Seifert matrices can be derived from Dehn presentations.
Core groups are link invariants defined by arc or region presentations.
problem Defining link invariants using different presentations of arcs and regions.
method Introducing core groups as link invariants defined by presentations involving arcs or regions, and extending these to virtual link diagrams.
result Properties of core groups and their extensions to virtual link diagrams are discussed.
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 G on vertices X={x1,…,xk}, where each oriented edge is labeled by a word in X±1. 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.
problem Representing split and composite links in a clear, recognizable way.
method Pocket and flip moves to transform link presentations.
result Pocket moves are the only obstruction to representing split links by split plat presentations.
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 Sd. 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.
problem Finite presentations for mapping class groups of non-orientable surfaces.
method Using Stukow's finite presentation and Birman exact sequences.
result An infinite presentation for the twist subgroup of the mapping class group of a compact non-orientable surface.
Paper extends method of presenting surface-links to immersed surface-links.
problem Presenting immersed surface-links in 4-space.
method Use marked graph diagrams with moves preserving isotopy classes.
result Extended method for presenting immersed surface-links.
Paper proves vanishing homology groups for certain hyperbolic groups.
problem Understanding homology groups of specific hyperbolic groups.
method Using twisted Wirtinger presentations to prove homology group vanishing.
result Second homology groups vanish for certain Gromov hyperbolic groups.
New 3-manifold spines with unique Whitehead graphs identified.
problem Identifying new 3-manifold spines with specific Whitehead graph types.
method Cyclic presentations and conditions on parameters to show spines of 3-manifolds.
result Examples of 3-manifolds with Whitehead graphs of specific types.
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.
Computes presentations for specific arithmetic hyperbolic lattices.
problem Computing presentations for cusped arithmetic hyperbolic lattices.
method Applying Macbeath's classical result to invariant horoball covers.
result Computations for specific groups like Picard modular and quaternion hyperbolic.
Defines a cell complex for even spin mapping class group.
problem No specific problem stated; focuses on definition.
method Defines a cell complex with an action of the even spin mapping class group.
result Obtains a finite presentation of the even spin mapping class group.
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.
The paper provides infinite presentations for surface groups.
problem Understanding fundamental groups of surfaces.
method Infinite presentations of fundamental groups using simple loops.
result Infinite presentations for fundamental groups of surfaces and non-orientable surfaces.
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…
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.
The study aims to prevent unfair content presentation in recommender systems.
problem Over- and under-presentation of content leads to biased user preference estimates.
method Two models are considered: one that ignores systematic and limited exposure, and another that conditions on limited exposure.
result Ignoring systematic presentations overestimates promoted options and underestimates censored alternatives.
Alternative proof of Dynnikov's three-page index for torus links.
problem Proving the existence of a three-page presentation for any link.
method Provided an alternative proof and defined the three-page index.
result Determined exact three-page indices for several 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.
problem Presented a simple group presentation for the pure cactus group of degree four.
method Action on hyperbolic plane, Dirichlet polygon construction.
result Isomorphic to the fundamental group of connected sum of five real projective planes.
New findings on stable commutator lengths in recursively presented groups.
problem Understanding stable commutator lengths in recursively presented groups.
method Analyzing recursively presented groups and infinitely presented small cancellation groups.
result All non-negative algebraic or computable numbers are in the set of stable commutator lengths.
Impact of chosen behavioural factors on imprecision of present value is discussed here. The formal model of behavioural present value is offered as a result of this discussion. Behavioural present value is described here by fuzzy set. These considerations were illustrated by means of extensive numerical case study. Fin…
In the present paper, we describe new approaches for constructing virtual knot invariants. The main background of this paper comes from formulating and bringing together the ideas of biquandle (Kauffman and Radford) the virtual quandle (Manturov), the ideas of quaternion biquandles by Roger Fenn and Andrew Bartholomew,…
Paper explores link and plat presentations, showing equivalence under bridge isotopy.
problem Relationship between links in bridge position and plat presentations.
method Established equivalence between Hilden double coset classes of plat presentations and bridge positions up to isotopy.
result Revealed equivalence of Hilden double coset classes for n-bridge unknot and torus knots in plat position.
The paper presents presentations for Euclidean Picard modular groups.
problem Finding presentations for arithmetic non-cocompact lattices in Euclidean domains.
method Applying Macbeath's theorem to a Γ-invariant covering by horoballs.
result Presentations for Picard modular groups with d=2,11 are obtained.
Finite presentations for skein algebras linked to gauge field theory.
problem Understanding finite presentations for skein algebras and their relationship to gauge field theory.
method Provided finite presentations and deduced properties of stated skein algebras.
result Stated skein algebras are Koszul and isomorphic to quantum moduli algebras in gauge field theory.
In this paper, a method is given to calculate the Jones polynomial of the 6-plat presentations of knots by using a representation of the braid group B6 into a group of 5×5 matrices. We also can calculate the Jones polynomial of the 2n-plat presentations of knots by generalizing the method for the …
Study infinite group presentations and their Dehn functions.
problem Behavior of Dehn functions for groups with infinite sets of relators.
method Analyzing Dehn functions for specific groups and presentations.
result Discovered continuum many distinct growth types of Dehn functions for the group Z^2.