Search

In this paper, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary. [ABSTRACT FROM AUTHOR]

A spiral knot or link diagram (introduced in [C. Adams, R. Hudson, R. Morrison, W. George, L. Starkston, S. Taylor and O. Turanova, The spiral index of knots, Math. Proc. Cambridge Philos. Soc. 149(2) (2010) 297–315]) is an oriented knot or link diagram where, when traversing through the planar diagram, the curvature does not change sign. An oriented knot or link type is called curly if it admits a spiral diagram with fewer maxima than the braid index of that knot or link type. In this paper, we exhibit families of curly knots and links. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the embedded homology groups of hypergraphs initially given in [S. Bressan, J. Li, S. Ren and J. Wu, The embedded homology of hypergraphs and applications, Asian J. Math. 23(3) (2019) 479–500] and study the relative embedded homology groups of hypergraph pairs. We prove some long exact sequences as well as a Mayer–Vietoris sequence for the relative embedded homology groups of hypergraph pairs. Moreover, we briefly discuss the two-dimensional persistence for the relative embedded homology groups of hypergraph pairs. [ABSTRACT FROM AUTHOR]

This paper defines a new operation through extending the idea of the 0-dimensional crossing change and Shimizu's 2-dimensional region crossing change [A. Shimizu, Region crossing change is an unknotting operation, J. Math. Soc. Jpn. 66(3) (2014) 693–708, doi:10.2969/jmsj/06630693] to a 1-dimensional version called the arc crossing change. We will also prove that the arc crossing change is an unknotting operation with the help of Gauss diagrams. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas has been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to present the seven invariants of the degree three (Proposition 4). We further give 2 3 Gauss diagram formulas of classical knots (Proposition 5). In particular, the Polyak–Viro Gauss diagram formula [M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not.1994 (1994) 445–453] is not a long virtual knot invariant; however, it is included in the list of 2 3 formulas. It has been unknown whether this formula would be available by arrow diagram calculus automatically. In consequence, as it relates to the conjecture of Goussarov-Polyak-Viro [Finite-type invariants of classical and virtual knots, Topology39 (2000) 1045–1068, Conjecture 3.C], for all the degree three finite type long virtual knot invariants, each Gauss diagram formula is represented as those of Vassiliev invariants of classical knots (Theorem 1). [ABSTRACT FROM AUTHOR]

A spiral knot or link diagram (introduced in [C. Adams, R. Hudson, Rachel, R. Morrison, W. George, L. Starkston, S. Taylor and O. Turanova, The spiral index of knots, Math. Proc. Cambridge Philos. Soc. 149(2) (2010) 297–315]) is an oriented knot or link diagram where, when traversing through the planar diagram, the curvature does not change sign. An oriented knot or link type is called curly if it admits a spiral diagram with fewer maxima than the braid index of that knot or link type. We prove that all 2-bridge knots and links with a braid index greater than three are curly. We also show that many alternating oriented Montesinos links are curly. [ABSTRACT FROM AUTHOR]