Narrow Results

We introduce the notion of the framed Thompson group, which can be seen as a categorification of the ordinary Thompson group, and we show how framed links can be obtained from elements of the framed Thompson group.

This text is a brief description of the Jones polynomial as it appears in the announcement in [9], together with Jones’ comments in the letter he addressed to Joan Birman on 31st May 1984 [8]. As much as possible I have followed the author’s initial notations. I add comments about the history of the events as I remember them, with the help of several testimonies of the “actors”. Among them: Didier Hatt-Arnold, Pierre de la Harpe, Cam Van Quach Hongler. It took me nontrivial efforts with my memories to get the chronology right (I hope so). The help of Eva Bayer was decisive. My warmest thanks to her. One may disagree with some of my statements which express a personal opinion. On purpose I have adopted a candid style.

The complex eikonal equation in the three space dimensions is considered. We show that apart from the recently found torus knots, this equation can also generate other topological configurations with a nontrivial value of the π₂(S²) index: braided open strings as well as hedgehogs. In particular, cylindric strings, i.e. string solutions located on a cylinder with a constant radius are found. Moreover, solutions describing strings lying on an arbitrary surface topologically equivalent to cylinder are presented. We discuss them in the context of the eikonal knots. The physical importance of the results originates in the fact that the eikonal knots have been recently used to approximate the Faddeev–Niemi hopfions.

In this paper we study unitary braid group representations associated with Majorana Fermions. Majorana Fermions are represented by Majorana operators, elements of a Clifford algebra. The paper recalls and proves a general result about braid group representations associated with Clifford algebras, and compares this result with the Ivanov braiding associated with Majorana operators. The paper generalizes observations of Kauffman and Lomonaco and of Mo-Lin Ge to show that certain strings of Majorana operators give rise to extraspecial 2-groups and to braiding representations of the Ivanov type.

We give an elementary construction of the Fibonacci model, a unitary braid group representation that is universal for quantum computation. This paper is dedicated to Professor C. N. Yang, on his 85th birthday.

In this paper, we give an explicit construction of dynamical systems (defined within a solid torus) containing any knot (or link) and arbitrarily knotted chaos. The first is achieved by expressing the knots in terms of braids, defining a system containing the braids and extending periodically to obtain a system naturally defined on a torus and which contains the given knotted trajectories. To get explicit differential equations for dynamical systems containing the braids, we will use a certain function to define a tube neighborhood of the braid. The second one, generating chaotic systems, is realized by modeling the Smale horseshoe.

We introduce a new approach to Vassiliev invariants. This approach deals with Vassiliev invariants directly on knots and does not make use of diagrams. We give a series of applications of this approach, (re)proving some new and known facts on Vassiliev invariants.

We show that Markov moves cannot be replaced by double Markov moves.

We study bifurcations of Voronoi diagrams on the plane. The generic bifurcations of Voronoi diagrams for moving points are classified into four types.

A braid has an associated family of Voronoi diagrams. If we admit only three types among four in the associated generic family of Voronoi diagrams, the braid type reduces to a braid with k half twists for some integer k.

We study automorphisms of surface braid groups and we compute the corresponding outer automorphism groups.

We initiate a new way to study presentations of the inverse braid monoid . Our method makes use of the monoid of all order-preserving partial permutations on an n-element chain. As an application we derive the Easdown–Lavers (2004) presentation of , and we also obtain new presentations for the symmetric inverse monoid and the monoid of all order-preserving partial braids with at most n strings.

We prove that the Murasugi–Przytycki index of the link graph determines an upper bound for the number of reducing operations that can be performed on a link diagram to reduce the number of Seifert circles.

A Chebyshev knot is a knot which admits a parametrization of the form x(t) = T_a(t); y(t) = T_b(t); z(t) = T_c(t + ϕ), where a, b, c integers, T_n(t) is the Chebyshev polynomial of degree n, and φ ∈ R. Chebyshev knots are non-compact analogues of the classical Lissajous knots. We show that there are infinitely many Chebyshev knots with φ = 0. We also show that every knot is a Chebyshev knot.

We study knots in 𝕊³ obtained by the intersection of a minimal surface in ℝ⁴ with a small 3-sphere centered at a branch point. We construct new examples of minimal knots. In particular we show the existence of non-fibered minimal knots. We show that simple minimal knots are either reversible or fully amphicheiral; this yields an obstruction for a given knot to be a simple minimal knot. Properties and invariants of these knots such as the algebraic crossing number of a braid representative and the Alexander polynomial are studied.

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 are non-negative). We also show that this result cannot be extended to a higher number of strands, as we provide a strongly quasipositive braid whose closure has non-positive Conway polynomial.

Let $M$ be a closed, connected and oriented surface of genus $g\ge 1$. In this work, we prove that the homotopy string links over$M$, namely, ${\widehat{PB}}_n(M)$, are bi-orderable, i.e. there is an (explicit) left and right order for such group.

We study the degree of polynomial representations of knots. We obtain the lexicographic degree for two-bridge torus knots and generalized twist knots. The proof uses the braid theoretical method developed by Orevkov to study real plane curves, combined with previous results from [Chebyshev diagrams for two-bridge knots, Geom. Dedicata150 (2010) 405–425; E. Brugallé, P.-V. Koseleff, D. Pecker, Untangling trigonal diagrams, to appear in J. Knot Theory and its Ramifications]. We also give a sharp lower bound for the lexicographic degree of any knot, using real polynomial curves properties.

In this paper, we study unitary braid group representations associated with Majorana fermions. Majorana fermions are represented by Majorana operators, elements of a Clifford algebra. The paper proves a general result about braid group representations associated with Clifford algebras and compares this result with the Ivanov braiding associated with Majorana operators and with other braiding representations associated with Majorana fermions such as the Fibonacci model for universal topological quantum computing.

In the present paper, a new representation of the virtual braid group ${VB}_n$ into the automorphism group of free product of the free group and free abelian group is constructed. This representation generalizes the previously constructed ones. The fact that the previously known representations are not faithful for $n\ge 4$ is verified. Using representations of ${VB}_n$, a virtual link group is defined. Also representations of the welded braid group ${WB}_n$ are constructed and the welded link group is defined.

This paper establishes that sutured annular Khovanov homology is not invariant for braid closures under axis-preserving mutations. This follows from an explicit relationship between sutured annular Khovanov homology and the classical Burau representation for braid closures.