Related papers: Links arising from braid monodromy factorizations

We construct certain monoids, called tied monoids. These monoids result to be semidirect products finitely presented and commonly built from braid groups and their relatives acting on monoids of set partitions. The nature of our monoids…

Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple…

We introduce the notion of a non-degenerate solution of the braid equation on the incidence coalgebra of a locally finite order. Each one of these solutions induce by restriction a non-degenerate set theoretic solution over the underlying…

In this paper we introduce the tied links, i.e. ordinary links provided with some ties between strands. The motivation for introducing such objects originates from a diagrammatical interpretation of the defining generators of the so-called…

We show that there are only finitely many homogeneous links whose Conway polynomial has any given degree. Using this we give an example of an inhomogeneous, fibred knot. Secondly, we show how to compute the monodromy of a homogeneous link…

Localized deformation patterns are a common motif in morphogenesis and are increasingly finding widespread applications in materials science, for instance as memory devices. Here we describe the emergence of spatially localized deformations…

In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic…

These are the notes of the lectures delivered by the author at CIME in June 2018. The main purpose of the notes is to provide an overview of the techniques used in the construction of the triply graded link homology. The homology is space…

For an oriented surface link $S$, we can take a satellite construction called a 2-dimensional braid over $S$, which is a surface link in the form of a covering over $S$. We demonstrate that 2-dimensional braids over surface links are useful…

We study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. a network consisting of nodes and edges. Our proposed set of generators consists of exchanges of pairs of particles on junctions of…

In this paper, we study the geometry of trisections on certain rational elliptic surfaces. We utilize Mumford representations of semi-reduced divisors in order to construct trisections and related plane curves with interesting properties…

Any link that is the closure of a positive braid has a natural Legendrian representative. These were introduced in an earlier paper, where their Chekanov--Eliashberg contact homology was also evaluated. In this paper we re-phrase and…

Knots and links which are closed 3-braids are a very special class. Like 2-bridge knots and links, they are simple enough to admit a complete classification. At the same time they are rich enough to serve as a source of examples on which,…

Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants, stable on…

We study projective structures on a surface having poles of prescribed orders. We obtain a monodromy map from a complex manifold parameterising such structures to the stack of framed $\mathrm{PGL}_2(\mathbb{C})$ local systems on the…

We study a wide range of homologically-defined representations of surface braid groups and of mapping class groups of surfaces, including the Lawrence-Bigelow representations of the classical braid groups. These representations naturally…

The purpose of this work is to collect in one place available information on line arrangements known in the literature as braid, monomial, Ceva or Fermat arrangement. They have been studied for a long time and appeared recently in…

We list all the possible fundamental groups of the complements of real conic-line arrangements with two conics which are tangent to each other at two points, with up to two additional lines. For the computations we use the topological local…

This is an expository article on diagrammatic representations of knots and links in various settings via braids.

The closure of a braid in a closed orientable surface $\Sigma$ is a link in $\Sigma\times S^1$. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids),…