Related papers: Trisected Rainbows and Braids
We introduce the concept of pseudo-trisections of smooth oriented compact 4-manifolds with boundary. The main feature of pseudo-trisections is that they have lower complexity than relative trisections for given 4-manifolds. We prove…
It is well known that the braid index of a link equals the minimum number of Seifert circles among all link diagrams representing it. For a link with a reduced alternating diagram $D$, $s(D)$, the number of Seifert circles in $D$, equals…
We prove that every smoothly embedded surface in a 4--manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4--manifold; that is, after isotopy, the surface meets components of the trisection in…
In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index $b(L)$ and arc index…
We use grid diagrams to present a unified picture of braids, Legendrian knots, and transverse knots.
In this article, we propose the approach to procedural optimization of a neural network, based on the combination of information theory and braid theory. The network studied in the article implemented with the intersections between the…
A special class of braids, called woven, is introduced and it is shown that every conjugation class of the braid group contains woven braids. In consequence, links can be presented as plats or closures of woven braids. Restricting on knots,…
The surface singular braid monoid corresponds to marked graph diagrams of knotted surfaces in braid form. In a quest to resolve linearity problem for this monoid, we will show that if it is defined on at least two or at least three strands,…
Square grids play a pivotal role in Robertson and Seymour's work on graph minors as planar obstructions to small treewidth. We introduce a three-sided bramble in a plane graph called a net, which generalizes the standard bramble of crosses…
The contents of this 6-page paper have been subsumed into the 13-page paper, "A note on closed 3-braids", arXiv:0802.1072 [math.GT]. This paper is correct, but contains less information than the new one. The topological classification of…
We use the combinatorial techniques of graphs of intersection to study reducible Dehn surgeries on knots in the three-sphere. In particular, in the event that a reducible surgery on a knot K in the three-sphere of slope r produces a…
Let $G$ be a nontrivial connected graph with an edge-coloring $c: E(G)\rightarrow \{1,2,...,q\},$ $q \in \mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is a $rainbow tree$ if no two edges of $T$ receive the…
It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of…
Given a connected cobordism between two knots in the 3-sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has…
We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called \emph{pseudo-simultaneously flippable edges}. Such edges are related to the notion of convex…
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…
Given a knot or link in the form of plat closure of a braid, we describe an algorithm to obtain a braid representing the same knot or link with the standard closure, and vice-versa. We analyze the three cases of knots and links: in…
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…
Bridge multisections are combinatorial descriptions of surface links in 4-space using tuples of trivial tangles. They were introduced by Islambouli, Karimi, Lambert-Cole, and Meier to study curves in rational surfaces. In this paper, we…
We develop a technique for gluing relative trisection diagrams of $4$-manifolds with nonempty connected boundary to obtain trisection diagrams for closed $4$-manifolds. As an application, we describe a trisection of any closed $4$-manifold…