Related papers: A Laundry Surface for Braids

We prove that the pure braid groups on closed, orientable surfaces are bi-orderable, and that the pure braid groups on closed, non-orientable surfaces have generalized torsion, thus they are not bi-orderable.

Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such…

This paper is a survey of some properties of the braid groups and related groups that lead to questions on mapping class groups.

We examine spaces of connected tri-/univalent graphs subject to local relations which are motivated by the theory of Vassiliev invariants. It is shown that the behaviour of ladder-like subgraphs is strongly related to the parity of the…

A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we…

Alexander's and Markov's theorems state that any link type in $R^3$ is represented by a closed braid and that such representations are related by some elementary operations called Markov moves. We generalize the notion of a braid to that in…

We show that the link cobordism maps defined by the author are graded and satisfy a grading change formula. Using the grading change formula, we prove a new bound for $\Upsilon_K(t)$ for knot cobordisms in negative definite 4-manifolds. As…

A knot (or link) diagram is said to be everywhere equivalent if all the diagrams obtained by switching one crossing represent the same knot (or link). We classify such diagrams of a closed 3-braid.

In this survey paper we present the $L$--moves between braids and how they can adapt and serve for establishing and proving braid equivalence theorems for various diagrammatic settings, such as for classical knots, for knots in knot…

Twisted links are a generalization of classical links and correspond to stably equivalence classes of links in thickened surfaces. In this paper we introduce twisted intersection colorings of a diagram and construct two invariants of a…

We describe a method for generating minimal hard prime surface-link diagrams. We extend the known examples of minimal hard prime classical unknot and unlink diagrams up to three components and generate figures of all minimal hard prime…

We introduce and define "oriented framed measured lamination links" in a 3-manifold $M$. These generalize oriented framed links in 3-manifolds, and are confined to 2-dimensional improperly embedded subsurfaces of the 3-manifold. Just as…

A branched covering map of surfaces induces a map in the opposite direction between their arc complexes. We represent a branched covering map combinatorially using what we call a lifting picture, and use this representation to computably…

Using a super scalar field representation of the chiral vertex operators we develop a general method of calculating braiding matrices for all types of N=1 super-conformal 4-point blocks involving Ramond external weights. We give explicit…

Purpose of the Conference article, intended for a wider audience, is to introduce concepts and techniques used by Bronislaw Wajnryb and the author in order to show the diffeomorphism of certain elementary algebraic surfaces, called ABC…

We show that 3-braid links with given (non-zero) Alexander or Jones polynomial are finitely many, and can be effectively determined. We classify among closed 3-braids strongly quasipositive and fibered ones, and show that 3-braid links have…

We characterize sandwiched singularities in terms of their link in two different settings. We first prove that such singularities are precisely the normal surface singularities having self-similar non-archimedean links. We describe this…

The correspondence between 2-parameter families of oriented lines in ${\Bbb{R}}^3$ and surfaces in $T{\Bbb{P}}^1$ is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences…

Let $D$ be an oriented link diagram with the set of regions $\operatorname{r}_{D}$. We define a symmetric map (or matrix) $\operatorname{\tau}_{D}\colon\operatorname{r}_{D}\times \operatorname{r}_{D} \to \mathbb{Z}[x]$ that gives rise to an…

The use of complex networks as a modern approach to understanding the world and its dynamics is well-established in literature. The adjacency matrix, which provides a one-to-one representation of a complex network, can also yield several…