Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits

The purpose of this study is to establish the topological properties of three-dimensional manifolds which admit Morse – Smale flows without fixed points (non-singular or NMS-flows) and give examples of such manifolds that are not lens spaces. Despite the fact that it is known that any such manifold is a union of circular handles, their topology can be investigated additionally and refined in the case of a small number of orbits. For example, in the case of a flow with two non-twisted (having a tubular neighborhood homeomorphic to a solid torus) orbits, the topology of such manifolds is established exactly: any ambient manifold of an NMS-flow with two orbits is a lens space. Previously, it was believed that all prime manifolds admitting NMS-flows with at most three non-twisted orbits have the same topology. Methods. In this paper, we consider suspensions over Morse – Smale diffeomorphisms with three periodic orbits. These suspensions, in turn, are NMS-flows with three periodic trajectories. Universal coverings of the ambient manifolds of these flows and lens spaces are considered. Results. In this paper, we present a countable set of pairwise distinct simple 3-manifolds admitting NMS-flows with exactly three non-twisted orbits. Conclusion. From the results of this paper it follows that there is a countable set of pairwise distinct three-dimensional manifolds other than lens spaces, which refutes the previously published result that any simple orientable manifold admitting an NMS-flow with at most three orbits is lens space.

Tied links in various topological settings

Tied links in [Formula: see text] were introduced by Aicardi and Juyumaya as standard links in [Formula: see text] equipped with some non-embedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper, we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces [Formula: see text], in handlebodies of genus [Formula: see text], and in the complement of the [Formula: see text]-component unlink. We introduce the tied braid monoids [Formula: see text] by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we formulate and prove analogues of the Alexander and the Markov theorems for tied links in the 3-manifolds mentioned above. We also present an [Formula: see text]-move braid equivalence for tied braids and we discuss further research related to tied links in knot complements and c.c.o. 3-manifolds. The theory of tied links has potential use in some aspects of molecular biology.

Key Facts about Skyscrapers and Decomposition Space Theory

‘Key Facts about Skyscrapers and Decomposition Space Theory’ summarizes the input from Parts I and II needed for the remainder of the proof of the disc embedding theorem. Precise references to previous chapters are provided. This enables Part IV to be read independently from the previous parts, provided the reader is willing to accept the facts from Parts I and II summarized in this chapter. The listed facts include the shrinking of mixed, ramified Bing–Whitehead decompositions of the solid torus, the shrinking of null decompositions consisting of recursively starlike-equivalent sets, the ball to ball theorem, the skyscraper embedding theorem, and the collar adding lemma.

Replicable Rooms and Boundary Shrinkable Skyscrapers

‘Replicable Rooms and Boundary Shrinkable Skyscrapers’ points out precisely which properties of skyscrapers are required in the remainder of the proof of the disc embedding theorem. To achieve this, it introduces an abstraction of towers, known as buildings. The required properties for a generalized skyscraper include boundary shrinkability and replicability. The former allows the conclusion that the vertical boundary of a generalized skyscraper is a solid torus. Replicability ensures that any generalized skyscraper contains uncountably many other skyscrapers as subsets. Both of the above properties will be essential in the construction of the design in a subsequent chapter.

More 1-cocycles for classical knots

Let [Formula: see text] be the topological moduli space of long knots up to regular isotopy, and for any natural number [Formula: see text] let [Formula: see text] be the moduli space of all [Formula: see text]-cables of framed long knots which are twisted by a string link to a knot in the solid torus [Formula: see text]. We upgrade the Vassiliev invariant [Formula: see text] of a knot to an integer valued combinatorial 1-cocycle for [Formula: see text] by a very simple formula. This 1-cocycle depends on a natural number [Formula: see text] with [Formula: see text] as a parameter and we obtain a polynomial-valued 1-cocycle by taking the Lagrange interpolation polynomial with respect to the parameter. We show that it induces a non-trivial pairing on [Formula: see text] already for [Formula: see text].

Tied links in the solid torus

We introduce the concept of tied links in the solid torus, which generalizes naturally the concept of tied links in [Formula: see text] previously introduced by Aicardi and Juyumaya. We also define an invariant of these tied links by using skein relations, and we then recover this invariant by using Jones’ method over the bt-algebra of type [Formula: see text] and the Markov trace defined on this.