Journal of Knot Theory and Its Ramifications

A doodle is a collection of immersed circles without triple intersections in the 2-sphere. It was shown by the second author and Tayler that doodles induce commutator identities (identities amongst commutators) in a free group. In this paper, we observe this idea more closely by concentrating on doodles with proper noose systems and elementary commutator identities. In particular, we show that there is a bijection between cobordism classes of colored doodles and weak equivalence classes of elementary commutator identities.

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.

This is the story of my relationship with Patrick Dehornoy and his family, which began in 1996 and ended when Patrick passed away.

This paper, in homage to Patrick Dehornoy, tells the birth of Garside groups.

In addition to his extraordinary theoretical contributions, the computational aspects were a very important part of Patrick Dehornoy’s work. He developed algorithms and cared about their efficiency and their applications, at the same time as he was discovering the structure of the objects he was studying. In this note, I just recall some examples of this computational side of his research, that had a deep influence on me and many other researchers.

This paper recalls my first meeting with Patrick DeHornoy and its consequences.

Patrick Dehornoy’s mathematical legacy is impressive and it has been celebrated in other papers of this volume. However, I think that there is another important aspect of Patrick’s work that should be stressed, which is his involvement in scientific and managerial leadership in Mathematics. This text wants therefore to be a very personal memory of Patrick Dehornoy and of the impact that he had on my perspective on the relevance and importance of commitments to our community.

This paper in memory of Patrick Dehornoy (1952–2019) is an invitation to Garside theory for mainstream geometric group theorists interested in mapping class groups, curve complexes, and the geometry of Artin–Tits groups.

In this paper, we look at which Alexander and Markov theorems can be defined for generalized knot theories.

In the first of this series of papers, we looked at various generalized knot theories. In this follow up paper, we look at the theory of generalized braids and show how they are related to generalized knots. The twin groups are related to the theory of planar doodles and we prove a Markov-type theorem for them. An example and invariants are given to show that this is best possible. We also give conditions for when an exchange move is dependent on the other moves.