The paper finds minimal generating sets and abelianizes the quasitoric braid group.
problem Understanding the structure of quasitoric braids and their subgroup properties.
method Provided two minimal generating sets and determined the abelianization.
result Minimal generating sets and abelianization of the quasitoric braid group were determined.
Minimal braids found for quasipositive links, solving a Rudolph question.
problem Finding minimal braids for quasipositive links.
method Using quasipositive minimal braids to establish bounds and solve problems.
result Minimal braid representatives for quasipositive links exist and have specific properties.
An algorithm finds minimal generators for braid centralizers efficiently.
problem Finding minimal generators for braid centralizers efficiently.
method Using Garside theory, an algorithm computes minimal generators in quadratic time.
result The centralizer of a generic braid has a minimal set of generators with quadratic complexity.
Minimal complexes for two-strand braids defined directly.
problem Determining minimal complexes for braids.
method Explicit formulas derived from educated guesswork and reverse engineering.
result Construction of minimal complexes homotopy equivalent to Rickard complexes.
Adding a braid closure to a fibered knot makes a link ribbon concordance minimal.
problem Finding the minimal ribbon concordance for links in S3. method Link Floer homology combined with classical techniques.
result Any link L in S3 can be made ribbon concordance minimal by adding a single unknot. The paper finds braid representatives minimizing simple walks for knots.
problem Finding efficient braid representatives for knots.
method Developed methods to minimize the number of simple walks in braids.
result Computed the colored Jones polynomial for specific knots.
Minimal translation lengths for Torelli and pure braid groups on curve graphs are shown.
problem Understanding translation lengths of Torelli and pure braid groups on curve graphs.
method Analyzing asymptotic translation lengths of Torelli and pure braid groups on curve graphs.
result Minimal asymptotic translation lengths for Torelli and pure braid groups on curve graphs are shown to behave differently from their respective mapping class groups.
New invariant for braids found by adding imaginary generators.
problem Finding new invariants for classical braids.
method Constructing a monomorphism from pure braid group PBn into a larger group by adding imaginary generators. result Examples of minimal words in PBn are shown using the new invariant. Study on minimal stretching of surface braids with bounds.
problem Finding the least stretching of surface braids.
method Provided upper and lower bounds for minimal dilatation as a function of genus and punctures.
result Bounds on dilatation as genus increases and for a fixed number of punctures.
The article calculates the minimal model dimensions for classifying spaces of surface braid groups.
problem Classifying spaces for families of virtually abelian subgroups of surface braid groups.
method Analyzes the pure and full braid groups of surfaces with at least one boundary component or one puncture, proving analogous results for amenable subgroups.
result Minimal dimensions of models for classifying spaces are equal to the virtual cohomological dimension plus the rank of virtually abelian subgroups.
Study on braids and their impact on hyperbolic 3-manifolds' orderability.
problem Investigating the relationship between braids and the orderability of hyperbolic 3-manifolds.
method Examined faithful representations of braid groups on automorphism groups of free groups, focusing on bi-orderability.
result Found that the bi-orderability of hyperbolic 3-manifolds' fundamental groups is linked to the braids used.
New satellite knots found that can't be represented by positive braids with full twists.
problem Satellite knots that cannot be represented by positive braids with full twists.
method Analyzing positive minimal braids and their closures to find satellite knots.
result Infinitely many satellite knots with Lorenz patterns and companions are not Lorenz knots.
Braids can be represented geometrically as laminations of punctured disks. The geometric complexity of a braid is the minimal complexity of a lamination that represents it, and tight laminations are representatives of minimal complexity. These laminations give rise to a normal form of braids, via a relaxation algorithm…
We give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations. In particular we show that positive braid knots may not have positive minimal (strand number) braid representations, giving a counterpart to results of Franks-Williams a…
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…
Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility, stability and Reidemeister moves. We show that virtual braids are in a bijective…
Positive braids minimize knot untangling steps.
problem Finding the minimum number of steps to untangle knots.
method Analyzing positive braids and their knot closures, comparing ascending number to unknotting number.
result Ascending number equals unknotting number for knots from positive braids.
Neural nets solve braid untangling up to length 20.
problem Untangling braids in knot theory and group theory.
method Feed-forward neural networks in reinforcement learning.
result Trained neural networks to untangle braids in minimal moves.
New bounds on knot genus for positive braids.
problem Understanding the genus difference in positive braids.
method Analyzing the Seifert genus and 4-genus with respect to the number of strands.
result Characterizing knots with specific genus differences by forbidden minors.
Study of Lorenz links and T-links, showing equivalence and unique presentations.
problem Understanding equivalence and uniqueness of T-link presentations for Lorenz links.
method Minimal-braid approach, V-link braids, T-link parameters, geometric type criteria.
result Explicit correspondence between V-links and T-links, criteria for equivalence and uniqueness.
We solved a conjecture about braid group quotients being alternating groups.
problem Understanding the smallest non-trivial quotients of braid group commutator subgroups.
method Proved the conjecture about alternating groups as quotients, showed minimal quotient maps.
result Proved conjecture about braid group quotients being alternating groups.
Minimal generating sets of Reidemeister moves identified and classified.
problem Classifying minimal generating sets of Reidemeister moves.
method Determined minimal generating sets, provided classifications, and identified candidates.
result 12 out of 16 candidates for minimal generating sets were proven minimal.
No minimal charts with exactly seven white vertices found.
problem Finding minimal charts with specific vertex counts.
method Investigating charts representing embedded surfaces in 4-space.
result No minimal chart with exactly seven white vertices exists.
Let Dn denote the n-punctured disk in the complex plane, where the punctures are on the real axis. An n-braid α is said to be \emph{reducible} if there exists an essential curve system $\C$ in Dn, called a \emph{reduction system} of α, such that $α*\C=\C$ where $α*\C$ denotes the action of the braid α o…
Braids can be represented geometrically as curve diagrams. The geometric complexity of a braid is the minimal complexity of a curve diagram representing it. We introduce and study the corresponding notion of geometric generating function. We compute explicitly the geometric generating function for the group of braids o…
New procedures connect braid charts, triplane diagrams, and braid movies for knotted surfaces.
problem Understanding the braid index and bridge index of knotted surfaces in 4D.
method Introducing rainbow diagrams and new procedures for passing among triplane diagrams, braid movies, and braid charts.
result Inequalities relating braid index and bridge index of 2-knots are obtained.
Study shows surprising cobordism distances between certain torus knots.
problem Determining cobordism distances between thin and thick torus knots.
method Analyzes locally flat cobordisms between torus knots with small and large braid indices.
result Surprising fact about torus knots as cross-sections of almost minimal cobordisms.
The Upsilon invariant bounds cobordisms between knots and their braid index.
problem Bounding cobordisms between knots and their braid index.
method Using Ozsváth, Stipsicz, and Szabó's Upsilon-invariant.
result Established inductive formulas for the Upsilon invariant of torus knots.
Given a knot diagram D, we construct a semi-threading circle for it which can be an axis of D as a closed braid depending on knot diagrams. In particular, we consider semi-threading circles for minimal diagrams of a knot with respect to overpasses which give us some information related to the braid index. By this n…
We suggest a new algorithm for finding a canonical representative of a given braid, and also for the harder problem of finding a σ1-consistent representative. We conjecture that the algorithm is quadratic-time. We present numerical evidence for this conjecture, and prove two results: (1) The algorithm terminates in …
The braid group's commutator subgroup is generated by two elements for n ≥ 7.
problem Generating the smallest possible generating sets for the commutator subgroup of braid groups.
method Analyzing specific cases of braid groups (n=4, 6, 5, 7+) to find generating sets of minimal size.
result For n ≥ 7, the commutator subgroup of the braid group is generated by two elements.
Paper proves bounds on knot indices using bisected vertex leveling of plane graphs.
problem Upper bounds on knot indices using plane graph embeddings.
method Bisected vertex leveling of plane graphs to prove upper bounds on braid index and arc index.
result Quadratic upper bound on minimal crossing number of delta diagrams.
We use some Lie group theory and Budney's unitarization of the Lawrence-Krammer representation, to prove that for generic parameters of definite form the image of the representation (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, …
Study of generalized knots and links, proving inequality involving crossing number and braid index.
problem Proving an inequality involving the minimal crossing number and braid index for generalized knots and links.
method Introducing generalized crossings and moves, proving inequality for generalized knots and links.
result Proved inequality involving total crossing number and braid index for generalized knots and links.
Choose any oriented link type X and closed braid representatives X[+], X[-] of X, where X[-] has minimal braid index among all closed braid representatives of X. The main result of this paper is a `Markov theorem without stabilization'. It asserts that there is a complexity function and a finite set of `templates' such…
Paper classifies Heegaard Floer minimal knots in sutured manifolds.
problem Classifying minimal knots in sutured manifolds.
method Uses Heegaard Floer homology.
result Agrees with instanton Floer homology classification when applicable.
New representations of loop braid group from higher gauge theory effects.
problem Understanding representations of loop braid group in higher gauge theory.
method Introducing W-bikoids and constructing representations from groupoid algebras.
result Candidate for higher quantum group and flux metamorphosis.
Study on bounds of knot untangling for specific types of knots.
problem Determining upper limits for knot untangling.
method Defined warping degree, examined diagrams combinatorially.
result Upper bounds for unknotting and region unknotting numbers.
We consider a hyperbolic surface bundle over the circle with the smallest known volume among hyperbolic manifolds having 3 cusps, so called "the magic manifold". We compute the entropy function on the fiber face of the unit ball with respect to the Thurston norm, determine homology classes whose representatives are gen…
Study shows fibred knots can't be untied with specific moves.
problem Untangling fibred knots with specific moves.
method Analyzes fibred knots and their untangling with $ar{t}_{2k}$-moves.
result Upper bound on two strand twist operations for untangling knots.
We consider the (pure) braid groups B_{n}(M) and P_{n}(M), where M is the 2-sphere S^2 or the real projective plane RP^2. We determine the minimal cardinality of (normal) generating sets X of these groups, first when there is no restriction on X, and secondly when X consists of elements of finite order. This improves o…
The (1,1)-length of a knot is a braid group invariant equaling its level number.
problem Finding a numerical invariant for knot positions.
method Using braid group on two points in the torus to describe and calculate the (1,1)-length. result The (1,1)-length equals the level number of a knot. A minimal knot is the intersection of a topologically embedded branched minimal disk in R4 C2 with a small sphere centered at the branch point. When the lowest order terms in each coordinate component of the embedding of the disk in C2 are enough to determine the knot type, we talk …
This paper studies minimal charts of a specific type to understand embedded surfaces in 4-space.
problem Investigating minimal charts of a specific type to understand embedded surfaces in 4-space.
method Analyzing charts of type (5,2) to find a minimal chart.
result Identified a minimal chart of type (5,2) representing an embedded surface in 4-space.
No minimal chart of type (4,3) exists in 4-space.
problem Existence of minimal charts of specific type in 4-space.
method Investigation of charts and their properties in 4-space.
result No minimal chart of type (4,3) exists.
Study on defect of Bennequin-Eliashberg inequality and its relation to minimal genus Bennequin surfaces.
problem Exploring the defect of the Bennequin-Eliashberg inequality and its relation to minimal genus Bennequin surfaces.
method Defined the defect δ(T) of the Bennequin-Eliashberg inequality for null-homologous transverse links in contact manifolds with open books. Studied relations between δ(T) and minimal genus Bennequin surfaces, particularly in disk open book cases.
result The Bennequin inequality is sharp if and only if it is the closure of a strongly quasipositive braid, as shown by the equality δ(T) = N for certain conditions.
New findings on Stein fillings and braids in 3D and 4D spaces.
problem Understanding Stein fillings and braids in contact 3-manifolds and 4-ball.
method Analyzing Stein fillings and Lefschetz fibrations, studying braids and their factorizations.
result Existence of infinitely many Stein fillings with distinct homology groups for certain contact 3-manifolds.
It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links. The Morton-Franks-Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the …