Multiflypes of rectangular diagrams of links

We introduce a new very large family of transformations of rectangular diagrams of links that preserve the isotopy class of the link. We provide an example when two diagrams of the same complexity are related by such a transformation and are not obtained from one another by any sequence of “simpler” moves not increasing the complexity of the diagram along the way.

Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

The vanishing cycles of curves in toric surfaces II

We resume the study initiated in [R. Crétois and L. Lang, The vanishing cycles of curves in toric surfaces, I, preprint (2017), arXiv:1701.00608]. For a generic curve [Formula: see text] in an ample linear system [Formula: see text] on a toric surface [Formula: see text], a vanishing cycle of [Formula: see text] is an isotopy class of simple closed curve that can be contracted to a point along a degeneration of [Formula: see text] to a nodal curve in [Formula: see text]. The obstructions that prevent a simple closed curve in [Formula: see text] from being a vanishing cycle are encoded by the adjoint line bundle [Formula: see text]. In this paper, we consider the linear systems carrying the two simplest types of obstruction. Geometrically, these obstructions manifest on [Formula: see text] respectively as an hyperelliptic involution and as a spin structure. In both cases, we determine all the vanishing cycles by investigating the associated monodromy maps, whose target space is the mapping class group [Formula: see text]. We show that the image of the monodromy is the subgroup of [Formula: see text] preserving respectively the hyperelliptic involution and the spin structure. The results obtained here support Conjecture [Formula: see text] in [R. Crétois and L. Lang, The vanishing cycles of curves in toric surfaces, I, preprint (2017), arXiv:1701.00608] aiming to describe all the vanishing cycles for any pair [Formula: see text].

Algorithmic simplification of knot diagrams: New moves and experiments

This paper has an experimental nature and contains no new theorems. We introduce certain moves for classical knot diagrams that for all examples we have tested them on give a monotonic complete simplification. A complete simplification of a knot diagram [Formula: see text] is a sequence of moves that transform [Formula: see text] into a diagram [Formula: see text] with the minimal possible number of crossings for the isotopy class of the knot represented by [Formula: see text]. The simplification is monotonic if the number of crossings never increases along the sequence. Our moves are certain [Formula: see text] generalizing the classical Reidemeister moves [Formula: see text], and another one [Formula: see text] (together with a variant [Formula: see text]) aimed at detecting whether a knot diagram can be viewed as a connected sum of two easier ones. We present an accurate description of the moves and several results of our implementation of the simplification procedure based on them, publicly available on the web.

Conjugacy invariants for Brouwer mapping classes

We give new tools for homotopy Brouwer theory. In particular, we describe a canonical reducing set called the set of walls, which splits the plane into maximal translation areas and irreducible areas. We then focus on Brouwer mapping classes relative to four orbits and describe them explicitly by adding a tangle to Handel’s diagram and to the set of walls. This is essentially an isotopy class of simple closed curves in the cylinder minus two points.

LOCAL STRUCTURE OF IDEAL KNOTS, II CONSTANT CURVATURE CASE

The thickness, NIR (K) of a knot or link K is defined to be the radius of the largest open solid tube one can put around the curve without any self intersections of the normal discs, which is also known as the normal injectivity radius of K. For C1,1 curves K, [Formula: see text], where κ(K) is the generalized curvature, and the double critical self distance DCSD (K) is the shortest length of the segments perpendicular to K at both end points. The knots and links in ideal shapes (or tight knots or links) belong to the minima of ropelength = length/thickness within a fixed isotopy class. In this article, we prove that NIR (K) = ½ DCSC (K), for every relative minimum K of ropelength in Rn for certain dimensions n, including n = 3.

KNOTS THAT HAVE MORE THAN TWO ACCIDENTAL SLOPES

An accidental surface in the exterior of a knot in the 3-sphere is a closed essential surface for which there is an annulus in the knot exterior X connecting a loop in the surface and a nontrivial loop in ∂X, the peripheral torus of the knot. The isotopy class of the loop in ∂X is called an accidental slope; each accidental surface has a unique accidental slope. It is known that accidental slopes are integral or 1/0, and there is a knot with two accidental slopes 0 and 1/0. We show that for any integer m≥0, there is a hyperbolic knot which has m+1 accidental surfaces with accidental slopes 0,1,…,m.

EXISTENCE OF IDEAL KNOTS

Ideal knots are curves are that maximize the scale invariant ratio of thickness to length. Here we present a simple argument to establish the existence of ideal knots for each knot type and each isotopy class and show that they are C1,1 curves.

COURBES ALGÉBRIQUES RÉELLES ET COURBES FLEXIBLES SUR LES SURFACES RÉGLÉES DE BASE [Copf ]P1

A first aim of this paper is to answer, in the case of real ruled surfaces $X_l$ of base $\mathbb{C}P^1$, with $l \geq 2$, a question of V. A. Rokhlin: is it true that the equivariant isotopy class does not suffice to distinguish the connected components of the space of smooth real algebraic curves of $X_l$? A second aim is to prove that there exist in these surfaces some real schemes realized by real flexible curves but not by smooth real algebraic curves. These two results of real algebraic geometry are deduced from the following comparison theorem: when $m = l + 2k$, with $k > 0$, the discriminants of the surface $X_m$ are deduced from those of the surface $X_l$ via weighted homotheties. All these results are obtained from a study of a deformation of ruled surfaces.The paper is written in French.2000 Mathematical Subject Classification: 14H10, 14J26, 14P25.