arXiv research

Proves a quasi-order on fibre surfaces of positive braid links.

In this paper we clarify the relationship between ribbon surfaces of Legendrian graphs and quasipositive diagrams by using certain fence diagrams. As an application, we give an alternative proof of a theorem concerning a relationship between quasipositive fiber surfaces and contact structures on the 3-sphere. We also a…

We prove that any link admitting a diagram with a single negative crossing is strongly quasipositive. This answers a question of Stoimenow's in the (strong) positive. As a second main result, we give simple and complete characterizations of link diagrams with quasipositive canonical surface (the surface produced by Sei…

Study shows quasipositive fiber surfaces can't be well-ordered.

A necessary and sufficient condition for an oriented pretzel surface to be quasipositive yields an estimate for the slice genus of the boundary of an arbitrary oriented pretzel surface.

This article was originally published in Topology 31 (1992). The present hyperTeXed redaction corrects a few typographical errors and updates the references.

We study the generalization of quasipositive links from the three-sphere to arbitrary closed, orientable three-manifolds. Our main result shows that the boundary of any smooth, properly embedded complex curve in a Stein domain is a quasipositive link. This generalizes a result due to Boileau and Orevkov, and it provide…

The aim of this article is to give a characterization of strongly quasipositive quasi-alternating links and detect new classes of strongly quasipositive Montesinos links and non-strongly quasipositive Montesinos links. In this direction, we show that, if $L$ is an oriented quasi-alternating link with a quasi-alternatin…

Study non-fibered links' relation to tight contact structures.

Is any positive knot the closure of a positive braid? No. But if we consider positivity in terms of the generators of the braid group due to Birman, Ko and Lee, then the answer is yes. In this paper we prove that the same occurs when considering homogeneity. In the way we prove that the plumbing of two surfaces is a BK…

The paper finds a new lower bound on the genus of surfaces in indefinite 4-manifolds.

We define the Thurston-Bennequin polytope of a two-component link as the convex hull of all pairs of integers that arise as framings of a Legendrian representative. The main result of this paper is a description of the Thurston-Bennequin polytope for two-bridge links. As an application, we construct non-quasipositive s…

For a null-homologous transverse link $\mathcal T$ in a general contact manifold with an open book, we explore strongly quasipositive braids and Bennequin surfaces. We define the defect $δ(\mathcal T)$ of the Bennequin-Eliashberg inequality. We study relations between $δ(\mathcal T)$ and minimal genus Bennequin surface…

Proves alternating quasipositive links are positive and strongly quasipositive.

The study connects knot Floer homology and Lagrangian cobordisms.

Let S(D) be the surface produced by applying Seifert's algorithm to the oriented link diagram D. I prove that if D has no negative crossings then S(D) is a quasipositive Seifert surface, that is, S(D) embeds incompressibly on a fiber surface plumbed from positive Hopf annuli. This result, combined with the truth of the…

We show that every quasipositive link has a quasipositive minimal braid representative, partially resolving a question posed by Orevkov. These quasipositive minimal braids are used to show that the maximal self-linking number of a quasipositive link is bounded below by the negative of the minimal braid index, with equa…

Homogeneous quasipositive links are positive if their Seifert circles match braid index.

Links can be transformed into many others using a specific operation.

We discuss relations among various positivities of knots and links, such as strong quasipositivity and quasipositivity. We give several pieces of supporting evidence for conjectural statements concerning these positivities and the defect of Bennequin inequality. Finally, we determine strong quasipositivity and quasipos…

We prove that the connected sum of two links is quasipositive if and onlyif each summand is quasipositive. The prove is based on the filling disk technique

Simple condition proves when 2-bridge knots are quasipositive.

It has been known that any Alexander polynomial of a knot can be realized by a quasipositive knot. As a consequence, the Alexander polynomial cannot detect quasipositivity. In this paper we prove a similar result about Vassiliev invariants: for any oriented knot K and any natural number n there exists a quasipositive k…

Short note on braid index and quasipositivity of certain pretzel knots.

We give a computational algorithm which decides if a braid is quasipositive or not. A braid is quasipositive if it's a product of conjuguates of generators. For this, we use the theory of Garside and the combinatorials properties of the Artin monoid.

Characterizes a subset of links using quasipositive and homogeneous properties.

Strongly quasipositive links are those links which can be seen as closures of positive braids in terms of band generators. In this paper we give a necessary condition for a link with braid index 3 to be strongly quasipositive, by proving that in that case it has positive Conway polynomial (that is, all its coefficients…

New cobordisms found between certain quasipositive knots.

Study negative band numbers in braids and links.

Classifies manifolds with quasipositive curvature.

We generalize an algorithm of Rudolph to establish that every link is topologically concordant to a strongly quasipositive link.

We prove that every knot in the 3-space bounds an embedded punctured Moebius band whose other boundary component is a quasipositive fibred knot.

In this paper we use artificial neural networks to predict and help compute the values of certain knot invariants. In particular, we show that neural networks are able to predict when a knot is quasipositive with a high degree of accuracy. Given a knot with unknown quasipositivity we use these predictions to identify b…

We use the Birman-Ko-Lee presentation of the braid group to show that all closures of strongly quasipositive braids whose normal form contains a positive power of the dual Garside element $δ$ are fibered. We classify links which admit such a braid representative in geometric terms as boundaries of plumbings of positive…

We present a class of knots associated with labelled generic immersions of intervals into the plane and compute their Gordian numbers and 4-dimensional invariants. At least 10% of the knots in Rolfsen's table belong to this class of knots. We call them track knots. They are contained in the class of quasipositive knots…

In this paper we examine the relationship between various types of positivity for knots and the concodance invariant tau discovered by Ozsvath and Szabo and independently by Rasmussen. The main result shows that, for fibered knots, tau characterizes strong quasipositivity. This is quantified by the statement that for K…

We show that a link in an open book can be realized as a strongly quasipositive braid if and only if it bounds a Legendrian ribbon with respect to the associated contact structure. This generalizes a result due to Baader and Ishikawa for links in the three-sphere. We highlight some related techniques for determining wh…

We investigate the problem of characterising the family of strongly quasipositive links which have definite symmetrised Seifert forms and apply our results to the problem of determining when such a link can have an L-space cyclic branched cover. In particular, we show that if $δ_n = σ_1 σ_2 \ldots σ_{n-1}$ is the dual …

Let $L$ be a oriented link such that $Σ_n(L)$, the $n$-fold cyclic cover of $S^3$ branched over $L$, is an L-space for some $n \geq 2$. We show that if either $L$ is a strongly quasipositive link other than one with Alexander polynomial a multiple of $(t-1)^{2g(L) + (|L|-1)}$, or $L$ is a quasipositive link other than …

We study Khovanov homology classes which have state cycle representatives, and examine how they interact with Jacobsson homomorphisms and Lee's map $Φ$. As an application, we describe a general procedure, quasipositive modification, for constructing H-thick knots in rational Khovanov homology. Moreover, we show that sp…

We give a new characterization of symplectic surfaces in CP^2 via bridge trisections. Specifically, a minimal genus surface in CP^2 is smoothly isotopic to a symplectic surface if and only if it is smoothly isotopic to a surface in transverse bridge position. We discuss several potential applications, including the cla…

We present a simple combinatorial model for quasipositive surfaces and positive braids, based on embedded bipartite graphs. As a first application, we extend the well-known duality on standard diagrams of torus links to twisted torus links. We then introduce a combinatorial notion of adjacency for bipartite graph links…

Characterizes braid types and estimates twist coefficients.

[Original abstract (1992):] The modulus of quasipositivity q(K) of a knot K was introduced as a tool in the knot theory of complex plane curves, and can be applied to Legendrian knot theory in symplectic topology. It has also, however, a straightforward characterization in ordinary knot theory: q(K) is the supremum of …

It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomia…

Construct divide knots with specific genus properties.

Positive surgery on knots in weakly fillable 3-manifolds yields weakly fillable contact structures.

We characterize positive links in terms of strong quasipositivity, homogeneity and the value of Rasmussen, Beliakova and Wehrli's $s$-invariant. We also study almost positive links, in particular, determine the $s$-invariants of almost positive links. This result suggests that all almost positive links might be strongl…