Remarks on Formal Knot Theory (original) (raw)

Knot Polynomials: Myths and Reality

2011

This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, and Kaufman two-variable polynomial, Khovanov homology, factorizability of the polynomials, and knot primeness detection.

1 0 Ju l 2 01 1 KNOT POLYNOMIALS : MYTHS AND REALITY

2019

This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, and Kaufman two-variable polynomial, Khovanov homology, factorizability of the polynomials, and knot primeness detection.

Knot Theory: from Fox 3-colorings of links to Yang-Baxter homology and Khovanov homology

arXiv: Geometric Topology, 2016

This paper is an extended account of my "Introductory Plenary talk at Knots in Hellas 2016" conference We start from the short introduction to Knot Theory from the historical perspective, starting from Heraclas text (the first century AD), mentioning R.Llull (1232-1315), A.Kircher (1602-1680), Leibniz idea of Geometria Situs (1679), and J.B.Listing (student of Gauss) work of 1847. We spend some space on Ralph H. Fox (1913-1973) elementary introduction to diagram colorings (1956). In the second section we describe how Fox work was generalized to distributive colorings (racks and quandles) and eventually in the work of Jones and Turaev to link invariants via Yang-Baxter operators, here the importance of statistical mechanics to topology will be mentioned. Finally we describe recent developments which started with Mikhail Khovanov work on categorification of the Jones polynomial. By analogy to Khovanov homology we build homology of distributive structures (including homology ...

On Two Categorifications of the Arrow Polynomial for Virtual Knots

The Mathematics of Knots, 2011

Two categorifications are given for the arrow polynomial, an extension of the Kauffman bracket polynomial for virtual knots. The arrow polynomial extends the bracket polynomial to infinitely many variables, each variable corresponding to an integer arrow number calculated from each loop in an oriented state summation for the bracket. The categorifications are based on new gradings associated with these arrow numbers, and give homology theories associated with oriented virtual knots and links via extra structure on the Khovanov chain complex. Applications are given to the estimation of virtual crossing number and surface genus of virtual knots and links.

Khovanov Homology of Knots and Links

2019

As an aspiring pure mathematician, the University of Auckland’s summer research programme has provided me with an invaluable insight into the world of academia. My supervisor has exposed me to many fascinating and highly advanced areas of mathematics and, recognising my ambitions, he has expertly guided me to the forefront of research in knot invariants. I am very grateful for all of the opportunities I have had to interact with and learn from various faculty members, PhD students and other academics from institutions around the world whilst at the University of Auckland; in particular, for the opportunity to attend the conference, Character Varieties and Topological Quantum Field Theory, which introduced me to a number of active research areas related to knot theory and allowed me the chance to briefly discuss my research topic with New Zealand’s only Fields medalist and the creator of the Jones polynomial, Vaughan Jones.

Knot Theory and the Jones Polynomial

2018

In this paper we introduce the notion of a knot, knot equivalence, and their related terms. In turn, we establish that these terms correspond to the much simpler knot diagram, Reidemeister moves, and their related terms. It is then shown that working with these latter terms, one can construct a number of invariants that can be used to distinguish inequivalent knots. Lastly, the use of these invariants is applied in setting of molecular biology to study DNA supercoiling and recombination.

Compositions of Knots Using Alexander Polynomial

arXiv (Cornell University), 2023

Knot theory is the Mathematical study of knots. In this paper we have studied the Composition of two knots. Knot theory belongs to Mathematical field of Topology, where the topological concepts such as topological spaces, homeomorphisms, and homology are considered. We have studied the basics of knot theory, with special focus on Composition of knots, and knot determinants using Alexander Polynomials. And we have introduced the techniques to generalize the solution of composition of knots to present how knot determinants behave when we compose two knots.

Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots

Journal of Knot Theory and Its Ramifications, 2017

The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov–Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies as well to classical knots and classical knot cobordisms. We give an alternate formulation for the Manturov definition [34] of Khovanov homology [25], [26] for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We use this to show that a large class of virtual knots with unit Jones polynomial is non-classical, proving a conjecture in [20] and [10]. We then discuss the implications of the maps induced in the aforementioned theory to the universal Frobenius algebra [27] for virtual knots. Next we show how one can apply the Karoubi envelope approach of Bar-Natan and Morrison [3] on abstract link diagrams [17] with cross cuts to construct the canonical generators of the Khovan...