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Recently, the first named author defined a 2-parametric family of groups Gnk [V. O. Manturov, Non-reidemeister knot theory and its applications in dynamical systems, geometry and topology, preprint (2015), arXiv:1501.05208]. Those groups may be regarded as analogues of braid groups. Study of the connection between the groups Gnk and dynamical systems led to the discovery of the following fundamental principle: "If dynamical systems describing the motion of n particles possess a nice codimension one property governed by exactly kk particles, then these dynamical systems admit a topological invariant valued in Gnk". The Gnk groups have connections to different algebraic structures, Coxeter groups, Kirillov-Fomin algebras, and cluster algebras, to name three. Study of the Gnk groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the Gnk groups are reflections which make them similar to Coxeter groups and not to braid groups. Nevertheless, there are many ways to enhance Gnk groups to get rid of this 22-torsion. Later the first and the fourth named authors introduced and studied the second family of groups, denoted by Γnk, which are closely related to triangulations of manifolds. The spaces of triangulations of a given manifolds have been widely studied. The celebrated theorem of Pachner [P.L. homeomorphic manifolds are equivalent by elementary shellings, Europ. J. Combin.12(2) (1991) 129-145] says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. See also [I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, Boston, 1994); A. Nabutovsky, Fundamental group and contractible closed geodesics, Comm. Pure Appl. Math.49(12) (1996) 1257-1270]; the Γnk naturally appear when considering the set of triangulations with the fixed number of points. There are two ways of introducing the groups Γnk: the geometrical one, which depends on the metric, and the topological one. The second one can be thought of as a "braid group" of the manifold and, by definition, is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version. In this paper, we give a survey of the ideas lying in the foundation of the Gnk and Γnk theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories. [ABSTRACT FROM AUTHOR]

In this paper, we study representations of G n 3 -like groups. The group G n 3 itself appeared in works of the third named author on non-Reidemeister knot (and braid) theory. This group is closely related to dynamical systems of points and their invariants. Representations of G n 3 -like groups are useful both for the study of those groups themselves, and constructing invariants of knots and braids based on the G n 3 -like group structure. [ABSTRACT FROM AUTHOR]

A point in the (N , q) -torus knot in ℝ 3 goes q times along a vertical circle while this circle rotates N times around the vertical axis. In the Lissajous-toric knot K (N , q , p) , the point goes along a vertical Lissajous curve (parametrized by t ↦ (sin (q t + ϕ) , cos (p t + ψ))) while this curve rotates N times around the vertical axis. Such a knot has a natural braid representation B N , q , p which we investigate here. If gcd (q , p) = 1 , K (N , q , p) is ribbon; if gcd (q , p) = d > 1 , B N , q , p is the d th power of a braid which closes in a ribbon knot. We give an upper bound for the 4 -genus of K (N , q , p) in the spirit of the genus of torus knots; we also give examples of K (N , q , p) 's which are trivial knots. [ABSTRACT FROM AUTHOR]

Twisting a knot K in S³ along a disjoint unknot c produces a twist family of knots {Kn} indexed by the integers. We prove that if the ratio of the Seifert genus to the slice genus for knots in a twist family limits to 1, then the winding number of K about c equals either zero or the wrapping number. As a key application, if {Kn} or the mirror twist family {Kn} contains infinitely many tight fibered knots, then the latter must occur. This leads to the characterization that c is a braid axis of K if and only if both {Kn} and {Kn} each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for {Kn} to contain infinitely many L-space knots, and apply the characterization to prove that satellite L-space knots have braided patterns, which answers a question of both Baker-Moore and Hom in the positive. This result also implies an absence of essential Conway spheres for satellite L-space knots, which gives a partial answer to a conjecture of Lidman-Moore. [ABSTRACT FROM AUTHOR]

We study collections of planar curves that yield diagrams for all knots. In particular, we show that a very special class called potholder curves carries all knots. This has implications for realizing all knots and links as special types of meanders and braids. We also introduce and apply a method to compare the efficiency of various classes of curves that represent all knots. [ABSTRACT FROM AUTHOR]

The -Turk's head link is presented by the alternating diagram which is obtained from the standard diagram of the -torus link by crossing changes. In this paper, we show that for any integers and , the coefficients of in the Conway polynomials of the - and -Turk's head links coincide for and differ by sign for . We conjecture that this property holds for any . [ABSTRACT FROM AUTHOR]

In [V. O. Manturov, ], the second author defined the -free braid group with strands . These groups appear naturally as groups describing dynamical systems of particles in some 'general position'. Moreover, in [V. O. Manturov and I. M. Nikonov, J. Knot Theory Ramification 24 (2015) 1541009] the second author and Nikonov showed that is closely related to classical braids. The authors showed that there are homomorphisms from the pure braids group on strands to and and they defined homomorphisms from to the free products of . That is, there are invariants for pure free braids by and . On the other hand in [D. A. Fedoseev and V. O. Manturov, J. Knot Theory Ramification 24(13) (2015) 1541005, 12 pages] Fedoseev and the second author studied classical braids with addition structures: parity and points on each strands. The authors showed that the parity, which is an abstract structure, has geometric meaning - points on strands. In [S. Kim, arXiv:submit/1548032], the first author studied with parity and points. the author constructed a homomorphism from to the group with parity. In the present paper, we investigate the groups and extract new powerful invariants of classical braids from . In particular, these invariants allow one to distinguish the non-triviality of Brunnian braids. [ABSTRACT FROM AUTHOR]

In this paper, we introduce -braids and, more generally, -braids for an arbitrary group . They form a natural group-theoretic counterpart of -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. [ABSTRACT FROM AUTHOR]

In previous papers (see e.g. [V. O. Manturov, Parity in knot theory, Mat. Sbornik 201(5) (2010) 65-110]), the author realized the following principle for many knot theories: if a knot diagram is complicated enough then it reproduces itself, i.e. a subdiagram of any other diagram equivalent to it. This principle is realized by diagram-valued invariants [⋅] of knots such that [K] = K for K complicated enough. It turns out that in the case of free braids, the same principle can be realized in an unexpectedly easy way by a one-term invariant formula. [ABSTRACT FROM AUTHOR]

In this work we present a direct connection between Garside normal forms in the braid group B3 and their associated matrices in 2(ℕ) coming from the Nielsen-Thurston theory. As a consequence, we can give an upper and a lower bound for the dilatation factor of a rigid pseudo-Anosov three-strands braid with canonical length ℓ. [ABSTRACT FROM AUTHOR]

After summarizing from previous papers the definitions of the concepts associated with nets, i.e. triples of 6-transpositions in the Monster up to braiding, we give some results. [ABSTRACT FROM PUBLISHER]

We give an algorithm which decides the Nielsen-Thurston type of a given four-strand braid. The complexity of our algorithm is quadratic with respect to word length. The proof of its validity is based on a result which states that for a reducible 4-braid which is as short as possible within its conjugacy class (short in the sense of Garside), reducing curves surrounding three punctures must be round or almost round. As an application, we give a polynomial time solution to the conjugacy problem for non-pseudo-Anosov four-strand braids. [ABSTRACT FROM AUTHOR]

We define Dynnikov coordinates on virtual braid groups. We prove that they are faithful invariants of virtual 2-braids, and present evidence that they are also very powerful invariants for general virtual braids. [ABSTRACT FROM AUTHOR]

We give an upper bound for the dealternating number of a closed 3-braid. As applications, we determine the dealternating numbers, the alternation numbers and the Turaev genera of some closed positive 3-braids. We also show that there exist infinitely many positive knots with any dealternating number (or any alternation number) and any braid index. [ABSTRACT FROM AUTHOR]

We consider the problem of classification of links up to (2, 2)-moves. Our motivation comes from the theory of skein modules, more specifically from the skein module of S3 based on the deformation of (2, 2)-move. As it was proved in D-P-2, not every link can be reduced to a trivial link by (2, 2)-moves, for instance, the closure of (σ1σ2)6. In this paper, we classify 3-braids up to (2, 2)-moves and, we show how the Harikae–Nakanishi–Uchida conjecture can be modified to hold for closed 3-braids. As an important step in the classification we prove the conjecture for 2-algebraic links and classify (2, 2)-equivalence classes for links up to nine crossings. We also analyze an action of (2, 2)-move on Kei (involutive quandle) associated to a link diagram. We define Burnside Kei, Q(m, n), and ask for which values of m and n, is Q(m, n) finite. This question is motivated by classical Burnside problem. [ABSTRACT FROM AUTHOR]

In this paper we examine the relationship between various types of positivity for knots and the concordance invariant τ discovered by Ozsváth and Szabó and independently by Rasmussen. The main result shows that, for fibered knots, τ characterizes strong quasipositivity. This is quantified by the statement that for K fibered, τ(K) = g(K) if and only if K is strongly quasipositive. A corollary is that any knot admitting a lens space surgery or, more generally, an L-space surgery, is strongly quasipositive. In addition, we survey existing results regarding τ and forms of positivity and highlight several consequences concerning the types of knots which are (strongly) (quasi) positive. [ABSTRACT FROM AUTHOR]

The braid group Bn, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism rev : Bn →Bn, $v \mapsto \bar{v}$, defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation τ : x ↦ Δ-1xΔ by the generalized half-twist (Garside element). More generally, the involution rev is defined for all Artin groups (equipped with Artin's presentation) and the involution τ is defined for all Artin groups of finite type. A palindrome is an element invariant under rev. We study palindromes and palindromes invariant under τ in Artin groups of finite type. Our main results are the injectivity of the map $v \mapsto v\bar{v}$ in all finite-type Artin groups, the existence of a left-order compatible with rev for Artin groups of type A, B, D, and the existence of a decomposition for general palindromes. The uniqueness of the latter decomposition requires that the Artin groups carry a left-order. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For any knot K represented as an n-braid (n ≥ 3), we construct an infinite sequence of pairwise non-conjugate (n + 1)-braids {bm, m ∈ ℕ} representing K. [ABSTRACT FROM AUTHOR]

We use a refinement of an argument by Shinjo, and some study of the 3-strand Burau representation, to extend from knots to links her previous construction of infinite sequences of pairwise non-conjugate braids with the same closure of a non-minimal number of (and at least 4) strands. [ABSTRACT FROM AUTHOR]

Fix a straight line L in Euclidean 3-space and consider the fibration of the complement of L by half-planes. A generic knot K in the complement of L has neither fiber quadrisecants nor fiber extreme secants such that K touches the corresponding half-plane at 2 points. Both types of secants occur in generic isotopies of knots. We give lower bounds for the number of these fiber secants in all isotopies connecting given isotopic knots. The bounds are expressed in terms of invariants calculable in linear time with respect to the number of crossings. [ABSTRACT FROM AUTHOR]

This paper studies the biquandle as an invariant of virtual knots and links, analyzing its structure via composition of biquandle morphisms where elementary morphisms are associated with the crossings and virtual crossings in the diagram. [ABSTRACT FROM AUTHOR]

Given a knot theory (virtual, singular, knots in a 3-manifold etc.), there are deep relations between the diagrammatic knot equivalence in this theory, the braid structures and a corresponding braid equivalence. The L-moves between braids, due to their fundamental nature, may be adapted to any diagrammatic situation in order to formulate a corresponding braid equivalence. In this short paper, we discuss and compare various diagrammatic set-ups and results therein, in order to draw the underlying logic relating diagrammatic isotopy, braid structures, Markov theorems and L-move analogues. Finally, we apply our conclusions to singular braids. [ABSTRACT FROM AUTHOR]

We define a measure of "complexity" of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators Δij, which are Garside-like half-twists involving strings i through j, and by counting powered generators Δkij as log(ǀkǀ + 1) instead of simply ǀkǀ. The geometrical complexity is some natural measure of the amount of distortion of the n times punctured disk caused by a homeomorphism. Our main result is that the two notions of complexity are comparable. This gives rise to a new combinatorial model for the Teichmüller space of an n + 1 times punctured sphere. We also show how to recover a braid from its curve diagram in polynomial time. The key role in the proofs is played by a technique introduced by Agol, Hass, and Thurston. [ABSTRACT FROM AUTHOR]

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds. [ABSTRACT FROM AUTHOR]

The inverse braid monoid $\rscr{IB}_n$ 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 $\rscr{IB}_n$ the structure of an inverse monoid. We derive presentations of $\rscr{IB}_n$ by first using a groupoid presentation, and then deriving equivalent inverse monoid presentations. In particular, we offer a new derivation of the presentation for $\rscr{IB}_n$ that has been given by Easdown and Lavers. [ABSTRACT FROM AUTHOR]

The first author defined a well-order in the set of links by embedding it into a canonical well-ordered set of (integral) lattice points. He also gave elementary transformations among lattice points to enumerate the prime links in terms of lattice points under this order. In this paper, we add some new elementary transformations and explain how to enumerate the prime links. We show a table of the first 443 prime links arising from the lattice points of lengths up to 10 under this order. Our argument enables us to find 7 omissions and 1 overlap in Conway's table of prime links of 10 crossings. [ABSTRACT FROM AUTHOR]

A left orderable completely metrizable topological group is exhibited containing Artin's braid group on infinitely many strands. The group is the mapping class group (rel boundary) of the closed unit disk with a sequence of interior punctures converging to the boundary. This resolves an issue suggested by work of Dehornoy. [ABSTRACT FROM AUTHOR]

We propose a conjecture concerning 3-manifolds (which implies the ℚ conjecture of Myers') and construct a countable sequence of examples which supports it. [ABSTRACT FROM AUTHOR]

This brief report discusses some recent discoveries regarding Artin''s braid groups, concentrating on their orderability properties. [Copyright &y& Elsevier]

We show, using Mellit's recent results, that Kálmán's full twist formula for the HOMFLY polynomial can be generalized to a formula for superpolynomials in the case of positive toric braids. [ABSTRACT FROM AUTHOR]