Search

With the untimely passing of Patrick Dehornoy in September 2019, the world of mathematics lost a brilliant scholar who made profound contributions to set theory, algebra, topology, and even computer science and cryptography. And I lost a dear friend and a strong influence in the direction of my own research in mathematics. In this paper, I will concentrate on his remarkable discovery that the braid groups are left-orderable, and its consequences, and its strong influence on my own research. I’ll begin by describing how I learned of his work.

We show that for many classical knots one can find generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the (hyperbolic) knot $5_2$ and satellites of these knots., Comment: 7 pages, 8 figures

We study algebraic properties of groups of PL or smooth homeomorphisms of unit cubes in any dimension, fixed pointwise on the boundary, and more generally PL or smooth groups acting on manifolds and fixing pointwise a submanifold of codimension 1 (resp. codimension 2), and show that such groups are locally indicable (resp. circularly orderable). We also give many examples of interesting groups that can act, and discuss some other algebraic constraints that such groups must satisfy, including the fact that a group of PL homeomorphisms of the n-cube (fixed pointwise on the boundary) contains no elements that are more than exponentially distorted., 23 pages, 3 figures

It is a classical theorem that the free product of ordered groups is orderable. In this note we show that, using a method of G. Bergman, an ordering of the free product can be constructed in a functorial manner, in the category of ordered groups and order-preserving homomorphisms. With this functor interpreted as a tensor product this category becomes a tensor (or monoidal) category. Moreover, if $O(G)$ denotes the space of orderings of the group $G$ with the natural topology, then for fixed groups $F$ and $G$ our construction can be considered a function $O(F) \times O(G) \to O(F * G)$. We show that this function is continuous and injective. Similar results hold for left-ordered groups., Comment: 14 pages, 1 figure

This expository paper explores the interaction of group ordering with topological questions, especially in dimensions 2 and 3. Among the topics considered are surfaces, braid groups, 3-manifolds and their structures such as foliations and mappings between them. A final section explores currently ongoing research regarding spaces of homeomorphisms and their orderability properties. This is not meant to be a comprehensive survey, but rather just a taste of the rich relationship between topology and the theory of ordered groups.

It is well-known that there is a faithful representation of braid groups on automorphism groups of free groups, and it is also well-known that free groups are bi-orderable. We investigate which n-strand braids give rise to automorphisms which preserve some bi-ordering of the free group rank n. As a consequence of our work we find that of the two minimal volume hyperbolic 2-cusped orientable 3-manifolds, one has bi-orderable fundamental group whereas the other does not. We prove a similar result for the 1-cusped case, and have further results for more cusps. In addition, we study pseudo-Anosov braids and find that typically those with minimal dilatation are not order-preserving., Comment: 35 pages, 28 figures

We say that a knot $k_{1}$ in the $3$-sphere $1$-dominates another $k_{2}$ if there is a proper degree 1 map $E(k_{1})\to E(k_{2})$ between their exteriors, and write $k_{1}\ge k_{2}$. When $k_{1}\ge k_{2}$ but $k_{1}\ne k_{2}$ we write $k_{1}>k_{2}$. One expects in the latter eventuality that $k_{1}$ is more complicated. In this paper, we produce various sorts of evidence to support this philosophy.

We define a total ordering of the pure braid groups which is invariant under multiplication on both sides. This ordering is natural in several respects. Moreover, it well-orders the pure braids which are positive in the sense of Garside. The ordering is defined using a combination of Artin's combing technique and the Magnus expansion of free groups, and is explicit and algorithmic.By contrast, the full braid groups (on 3 or more strings) can be ordered in such a way as to be invariant on one side or the other, but not both simultaneously. Finally, we remark that the same type of ordering can be applied to the fundamental groups of certain complex hyperplane arrangements, a direct generalization of the pure braid groups.

It is known that knot groups are right-orderable, and that many of them are not bi-orderable. Here we show that certain bred knots in S 3 (or in a homology sphere) do have bi-orderable fundamental group. In particular, this holds for bred knots, such as 41, for which the Alexander polynomial has all roots real and positive. This is an application of the construction of orderings of groups, which are moreover invariant with respect to a certain automorphism.

This brief report discusses some recent discoveries regarding Artin's braid groups, concentrating on their orderability properties.

Groups which are locally indicable are also right-orderable, but not conversely. This paper considers a characterization of local indicability in right-ordered groups, the key concept being a property of right-ordered groups due to Conrad. Our methods answer a question regarding the Artin braid groups B n which are known to be right-orderable. The subgroups P n of pure braids enjoy an ordering which is invariant under multiplication on both sides, and it has been asked whether such an ordering of P n could extend to a right-invariant ordering of R n . We answer this in the negative. We also give another proof of a recent result of Linnell that for elementary amenable groups, the concepts of right-orderability and local indicability coincide.

We show how to use topological ideas, such as compactness, to establish orderability properties of infinite groups. A new application is to provide a left-ordering for the group of PL homeomorphisms of a connected surface with boundary which are fixed on at least one boundary component. This has been extended in recent joint work by the author and D. Calegari.

Any knot in a 3-dimensional homology sphere is homotopic to a knot with trivial Alexander polynomial.

We begin with the observation that the group algebras [Formula: see text] of Artin's braid groups have no zero divisors or nontrivial units. This follows from the recent discovery of Dehornoy that braids can be totally ordered by a relation < which is invariant under left multiplication. We then show that there is no ordering of Bn, n ≥ 3 which is simultaneously left and right invariant. Nevertheless, we argue that the subgroup of pure braids does possess a total ordering which is invariant on both sides. This follows from a general theorem regarding orderability of certain residually nilpotent groups. As an application, we show that the pure braid groups have no generalized torsion elements, although full braid groups do have such elements.

The classical braid groups of E. Artin form an ascending sequence B1 B2 . . . B1. These groups, and their representations, play an important roA le in several areas of mathematics and physics. The object of this note is to establish new algebraic information about this sequence of groups. In particular, we determine the normaliser and commensurator of the braid group Bn in Bm, n m 1. They are described in terms of the centraliser, which was recently characterised in [FRZ], and this leads to explicit generators and relations presenting these subgroups. In Sect. 5 we characterise the commensurator of Bn as a certain stabilizer, under the identi®cation of Bm as the mapping class group of the m-punctured 2-dimensional disk. This identi®cation gives rise to in®nitely many natural ``geometric'' inclusions of Bn in Bm, which we demonstrate to be mutually incommensurable, although they are all conjugate. Indeed, this point of view a€ords a simple description of the centralisers, normalisers and commensurators of all the geometric braid subgroups (Theorem 5.3). In Sect. 6 we show that the action of Bm, as the mapping class group, upon appropriate sets of curves, is a transitive and large action in the sense of Burger and de la Harpe [BH]. The paper concludes with some applications regarding unitary representations of the braid groups, and those induced by braid subgroups. Here is a summary of our principal results regarding the Bn sequence; the relevant de®nitions will follow.

We classify the compact 3-manifolds whose boundary is a union of 2-spheres, and which embed in T ×I ×I, where T is a triod and I the unit interval. This class is described explicitly as the set of punctured handlebodies. We also show that any 3-manifold in T × I × I embeds in a punctured handlebody.

We analyse earlier obtained numerical results for the knot complexity and the topological persistence length (i.e. the minimal number of steps on the lattice needed for the knot to be formed) with help of well known path integrals for semiflexible chains. In addition, we provide an estimate for the number of different knots having knot diagrams with exactly n crossings. Analytical results obtained are in satisfactory agreement with the available numerical data.

We discuss new methods of changing knot and link diagrams in such a way that their Jones polynomial is unaltered. It is similar to Conway's mutation, but applies to more complicated "tangles," and is performed simultaneously at multiple sites in the diagram — thus "global mutation." The method of achieving this can be regarded as communication via "messenger tangles."

We establish a necessary condition that an automorphism of a nontrivial finitely generated bi-orderable group can preserve a bi-ordering: at least one of its eigenvalues, suitably defined, must be real and positive. Applications are given to knot theory, spaces which fibre over the circle and to the Heegaard-Floer homology of surgery manifolds. In particular, we show that if a nontrivial fibred knot has bi-orderable knot group, then its Alexander polynomial has a positive real root. This implies that many specific knot groups are not bi-orderable. We also show that if the group of a nontrivial knot is bi-orderable, surgery on the knot cannot produce an $L$-space, as defined by Ozsv\'ath and Szab\'o., Comment: Minor changes from first version

We consider group orders and right-orders which are discrete, meaning there is a least element which is greater than the identity. We note that nonabelian free groups cannot be given discrete orders, although they do have right-orders which are discrete. More generally, we give necessary and sufficient conditions that a given orderable group can be endowed with a discrete order. In particular, every orderable group [math] embeds in a discretely orderable group. We also consider conditions on right-orderable groups to be discretely right-orderable. Finally, we discuss a number of illustrative examples involving discrete orderability, including the Artin braid groups and Bergman’s nonlocally-indicable right orderable groups.

Piecewise-linear (nonambient) isotopy of classical links may be regarded as link theory modulo knot theory. This note considers an adaptation of new (and old) polynomial link invariants to this theory, obtained simply by dividing a link's polynomial by the polynomials of the individual components. The resulting rational functions are effective in distinguishing isotopy classes of links, and in demonstrating that certain links are essentially knotted in the sense that every link in its isotopy class has a knotted component. We also establish geometric criteria for essential knotting of links.

We present examples showing that certain results on the invariance of link polynomials under generalized mutation are the best possible. They show, moreover, that this generalized mutation cannot be effected by a sequence of ordinary mutations. One of the examples also shows that the reduced Jones polynomial can be a more sensitive invariant than the Jones polynomial itself.