arXiv research

Article provides conditions for embedding braid groups into mapping class groups.

We construct an embedding of any right-angled Artin group $G(Δ)$ defined by a graph $Δ$ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of $Δ$. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid g…

The paper studies which branched covers can be lifted to braided embeddings.

An embedding of the m-times punctured disc into the n-times punctured disc, for n>m, yields an embedding of the braid group on m strands B_m into the braid group on n strands B_n, called a geometric embedding. The main example consists of adding n-m trivial strands to the right of each braid on m strands. We show that …

We show that after stabilizations of opposite parity and braid isotopy, any two braids in the same topological link type cobound embedded annuli. We use this to prove the generalized Jones conjecture relating the braid index and algebraic length of closed braids within a link type, following a reformulation of the prob…

Classic braids embed in virtual braids.

Formula for spectrum linking braid and bridge indices.

We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the non-orientable case. We show that these embeddings are non-geometric in the sense that t…

In this paper we study embeddings of contact manifolds using braidings of one manifold about another. In particular we show how to embed many contact 3-manifolds into the standard contact 5-sphere. We also show how to obstruct braidings of one manifold about another using contact geometry.

We prove that an arbitrary right-angled Artin group $G$ admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree. Consequently, $G$ admits quasi-isometric group embeddings into a pure braid group and into the area-preserving diffeomorphism groups of the 2--disk an…

Study local and global aspects of complex plane curve embeddings.

A large class of positive finite presentations of the braid groups is found and studied. It is shown that no presentations but known exceptions in this class have the property that equivalent braid words are also equivalent under positive relations.

We show that the "geometric models of matter" approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces. The anyon states arise through…

Study on planar graph braid groups' second homology.

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 …

Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d,n be positive integers, and let N be a d-fold covering space of M. We show that the covering map induces an embedding of the n-th braid group B_n(M) of M in the (dn)-th braid group B_{dn}(N) of N, and give sever…

Maximizes mixing efficiency in surface braids.

Study on embedding tree products into groups, distinguishing them.

In the present paper we give a new method for converting virtual knots and links to virtual braids. Indeed the braiding method given in this paper is quite general, and applies to all the categories in which braiding can be accomplished. We give a unifying topological interpretation of virtuals and flats (virtual strin…

New groups constructed from tree automorphisms, proving finiteness.

We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the $L^2$-norm metric); this extends resul…

In this paper we show that the singular braid monoid of an orientable surface can be embedded in a group. The proof is purely topological, making no use of the monoid presentation.

Spaces of circle embeddings in curved surfaces indexed by trees.

No minimal charts with exactly seven white vertices found.

We define the notion of a braided link cobordism in $S^3 \times [0,1]$, which generalizes Viro's closed surface braids in $\mathbb{R}^4$. We prove that any properly embedded oriented surface $W \subset S^3 \times [0,1]$ is isotopic to a surface in this special position, and that the isotopy can be taken rel boundary wh…

New braid group representations into mapping class groups are derived from disk coverings.

We study the rational permutation braids, that is the elements of an Artin-Tits group of spherical type which can be written $x^{-1} y$ where $x$ and $y$ are prefixes of the Garside element of the braid monoid. We give a geometric characterization of these braids in type $A_n$ and $B_n$ and then show that in spherical …

The singular braids with $n$ strands, $n \geq 3$, were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by $SG_n$. There has been another generalization of braid groups, denoted by $GVB_n$, $n \ge…

The paper studies submonoids of singular twisted virtual braids and their properties.

Developed algebraic theory of bonded braids, proving Markov theorem.

The topological underpinnings are presented for a new algorithm which answers the question: `Is a given knot the unknot?' The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that a knot is unknotted if and …

The article finds equivalence moves for links in specific manifolds using plat closure of braids.

This is a review article on the Bennequin-Birman-Menasco machinery for studying embedded incompressible surfaces in 3-space via their `braid foliations'. Two cases are investigated: case (1) The surface has non-empty boundary; the boundary is a knot or link which is represented as a closed braid, Case (2) The surface i…

Unified study of homological representations of mapping class groups.

We introduce new polynomial isotopy invariants for closed braids. They are constructed as polynomial valued {\em Gauss diagram 1-cocycles} evaluated on the full rotation of the closed braid $\hat β$ around the core of the corresponding solid torus. They can be calculated with polynomial complexity with respect to the b…

In this paper, we make use of the relations between the braid and mapping class groups of a compact, connected, non-orientable surface N without boundary and those of its orientable double covering S to study embeddings of these groups and their (virtual) cohomological dimensions. We first generalise results of Birman …

We generalize bridge trisections to surfaces in four-manifolds, linking them to braided links.

We show that the group cohomology of the diffeomorphisms of the disk with $n$ punctures has the cohomology of the braid group of $n$ strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the "standard" embedding $\mathrm{Br}_{2g+2}\hookrightarrow \…

This paper studies minimal charts of a specific type to understand embedded surfaces in 4-space.

We study simple branched coverings of degree d of the 2- and 3- dimensional sphere branched over oriented links. We demonstrate how to use braid charts to develop embeddings of these into $S^k \times D^2$ for $k=2,3 when$d=2,3$. This is an initial part of our study and represents the manuscript submitted to the RIMS w…

The abstract discusses braided surfaces and their characteristic maps, linking them to algebraic and geometric properties.

Study of bonded knots and braids with new algebraic models.

We classify 3-braids up to (2,2)-move equivalence and in particular we show how to adjust the Harikae-Nakanishi-Uchida conjecture so it holds for closed 3-braids. As important steps to classify 3-braids up to (2,2)-move equivalence we prove the conjecture for 2-algebraic links and classify (2,2)-equivalence classes for…

New surfaces in 4-ball constructed from knits, described by charts.

Let $φ: S^1\times D^2\to S^1$ be the natural projection. An oriented knot $K\hookrightarrow V = S^1\times D^2$ is called an almost closed braid if the restriction of $φ$ to K has exactly two (non-degenerate) critical points (and K is a closed braid if the restriction of $φ$ has no critical points at all). We introduce …

We study piecewise linear co-dimension two embeddings of closed oriented manifolds in Euclidean space, and show that any such embedding can always be isotoped to be a closed braid as long as the ambient dimension is at most five, extending results of Alexander (in ambient dimension three), and Viro and independently Ka…

No minimal chart of type (4,3) exists in 4-space.

A proof of non-existence of Lagrangian embeddings of the Klein bottle K in \CP^2 is given. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil on \CP^2 and study its monodromy. As the main technical tool, we develop the theory of mapping class groups, considered as quotients of special…