arXiv research

Paper proves any twisted link can be described as a unique twisted braid.

The paper extends knot theory to twisted virtual braids and links.

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

Study of decorated surfaces with vortices and their group structures.

Paper uses Long-Moody construction for new braid group representations.

Generators found for a specific subgroup of braid groups.

Classifies homomorphisms from braid groups to mapping class groups of nonorientable surfaces.

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

The Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials.

In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid groups. We also prove that half-twists satisfy a special root property which allows us to reduce the solution for the conjugacy problem in half-twists into the free group. Using this algorithm one is able to check conjugacy…

Evaluating the twisted Morita-Mumford classes bar(h)_p on the Artin braid group B_n, we give the stable algebraic independence of the bar(h)_p's on the automorphism group of the free group, Aut(F_n). This is sharper than the results we obtained by restricting them to the mapping class group.

Characterizes fractional Dehn twist coefficient and proves slice-Bennequin inequality.

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…

We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3-Calabi-Yau triangulated category. The motivation/application is that the…

The Helon model identifies Standard Model quarks and leptons with certain framed braids joined together at both ends by a connecting node (disk). These surfaces with boundary are called braided 3-belts (or simply belts). Twisting and braiding of ribbons composing braided 3-belts are interchangeable, and it was shown in…

By evaluating the Burau representation at t=-1, we obtain a symplectic representation of the braid group. We define the congruence subgroups of the braid group to be the preimages of the principal congruence subgroups of the symplectic group. Our main result is that the level four congruence subgroup of the braid group…

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic from the braid group to the mapping class group. We prove here that this map is trivial in stable homology with any trivial coefficients. In particular this proves an old conjecture of J. Harer…

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…

We consider the universal family $E_n^d$ of superelliptic curves: each curve $Σ_n^d$ in the family is a $d$-fold covering of the unit disk, totally ramified over a set $P$ of $n$ distinct points; $Σ_n^d\hookrightarrow E_n^d\to C_n$ is a fibre bundle, where $C_n$ is the configuration space of $n$ distinct points. We fin…

Study on singular twisted links and virtual braids, extending knot theory concepts.

We prove that the symplectic group $Sp(2n,\mathbb Z)$ and the mapping class group $Mod_{S}$ of a compact surface $S$ satisfy the $R_{\infty}$ property. We also show that $B_n(S)$, the full braid group on $n$-strings of a surface $S$, satisfies the $R_{\infty}$ property in the cases where $S$ is either the compact disk …

We are interested in the 3-Calabi-Yau categories $\mathcal{D}$ arising from quivers with potential associated to a triangulated marked surface $\mathbf{S}$ (without punctures). We prove that the spherical twist group ST of $\mathcal{D}$ is isomorphic to a subgroup (generated by braid twists) of the mapping class group …

The braid groups B_n can be defined as the mapping class group of the n-punctured disc. The Lawrence-Krammer representation of the braid group B_n is the induced action on a certain twisted second homology of the space of unordered pairs of points in the n-punctured disc. Recently, Daan Krammer showed that this is a fa…

Satellite links with many twists have simpler companions.

Study proves bridge and braid indices match for twist positive knots.

Positive braids with at least two twists form hyperbolic knots.

New satellite knots found that can't be represented by positive braids with full twists.

We interpret Coxeter's truncated braid groups in terms of Platonic solids.

Formula found for braid index of $n$-bridge braids.

Unified study of homological representations of mapping class groups.

Innovative warping labeling for twisted knots and braids.

Characterizes braid types and estimates twist coefficients.

A new presentation of a quotient of braid groups leads to a new type of Burnside group.

Study kernels of mapping class group representations on surface configuration spaces.

Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability for the family of groups. We show that stability also holds with both polynomial…

New link invariants derived from $L^2$-Burau maps of braids.

These are Lecture Notes of a course given by the author at the French-Spanish School "Tresses in Pau", held in Pau (France) in October 2009. It is basically an introduction to distinct approaches and techniques that can be used to show results in braid groups. Using these techniques we provide several proofs of well kn…

Polynomial representations found in surface braid and mapping class groups.

Researchers use Mellin-Barnes integrals to study trinomial equations and their braids.

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, …

Twisted links are obtained from a base link by starting with a $n$-braid representation, choosing several ($m$) adjacent strands, and applying one or more twists to the set. Various restrictions may be applied, e.g. the twists may be required to be positive or full twists, or the base braid may be required to have a ce…

The study connects twist positivity to L-space knots and concordance.

We characterize the fractional Dehn twist coefficient of a braid in terms of a slope of the homogenization of the Upsilon function, where Upsilon is the function-valued concordance homomorphism defined by Ozsváth, Stipsicz, and Szabó. We use this characterization to prove that $n$-braids with fractional Dehn twist coef…

Satellite links of fully positive braids are characterized.

New algebraic structure derived from Hopf algebra and Drinfel'd twist.

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 …

Calculates Dehn twist actions on conformal blocks for modular categories.

New model for Calabi-Yau-$\mathbb{X}$ categories using decorated marked surfaces.