Introductory Lectures on Knot Theory

The following sections are included:

• Contents

• Preface

In 2006 Habiro initiated a construction of generating functions for Witten–Reshetikhin–Turaev (WRT) invariants known as unified WRT invariants. In a series of papers together with Irmgard Bühler and Christian Blanchet we extended his construction to a larger class of 3–manifolds. The unified invariants provide a strong tool to study properties of the whole collection of WRT invariants, e.g. their integrality, and hence, their categorification.

In this paper we give a survey on ideas and techniques used in the construction of the unified invariants.

This article surveys many aspects of the theory of quandles which algebraically encode the Reidemeister moves. In addition to knot theory, quandles have found applications in other areas which are only mentioned in passing here. The main purpose is to give a short introduction to the subject and a guide to the applications that have been found thus far for quandle cocycle invariants.

This paper is an introductory survey of the combinatorial aspects of the Vassiliev theory of knot invariants following the lectures delivered at the Advanced School on Knot Theory and its Applications to Physics and Biology in the ICTP, Trieste (Italy), May 2009. The exposition is based on the forthcoming book [CDM], where the reader may find further details, examples, and developments.

We survey two of the many aspects of the standard braid order, namely its set theoretical roots, and the known connections with knot theory, including results by Netsvetaev, Malyutin, and Ito, and very recent work in progress by Fromentin and Gebhardt.

A traditional method of finding topological invariants of a knot in 3-dimensional euclidean space is to colour the arcs of a planar diagram and use this to extract invariants of the knot such as the fundamental group. This method can be extended in many ways to other kinds of codimension 2 embeddings. Moreover the new algebraic objects which arise quite naturally from these generalisations such as quandles, racks, biquandles etc are of interest in their own right. In this paper we shall give a brief overview of the methods and the new algebraic structures.

We consider triples (M; α, β), where M is a hyperbolic 3-manifold with boundary a disjoint union of tori, and α, β are distinct slopes on some boundary component such that the Dehn fillings M(α) and M(β) are not hyperbolic. Although there are infinitely many such (M; α, β)'s, we examine the question of whether they can all be obtained from a finite set by Dehn filling along some additional boundary components. If the distance Δ(α, β) between the slopes is 1 or 2 then this is not the case, but we show that it might be true if Δ(α, β) ≥ 3 and summarize what is known in this direction.

The present paper is a review of the current state of Graph-Link Theory (graph-links are closely related to homotopy classes of looped interlacement graphs): a theory suggested in,^1,2 see also,³ dealing with a generalisation of knots obtained by translating the Reidemeister moves for links into the language of intersection graphs of chord diagrams. In this paper we show how some methods of classical and virtual knot theory can be translated into the language of abstract graphs, and some theorems can be reproved and generalised to this graphical setting. We construct various invariants, prove certain minimality theorems and construct functorial mappings for graph-knots and graph-links. In this paper, we first show non-equivalence of some graph-links to virtual links.

We analyze relations between several knot invariants such as the unknotting (unlinking) number, adequacy, and quasi-alternating property, and the corresponding values obtained from (minimal crossing number) knot diagrams. In particular, we define the BJ-unlinking number, and compute it for various families of knots and links for which the unlinking number is unknown.

The following sections are included:

• Introduction

• The Culprit

• Rational Tangles, Rational Knots and Continued Fractions
  • The Tangle Fraction
  • Rational Knots and Continued Fractions

• The Return of the Culprit

• Collapsing to Unknots and Unlinks

• Continued Fractions, Convergents and Lots of Unknots

• Constructing Hard Unknots

• The Smallest Hard Unknots

• The Goeritz Unknot

• Recalcitrance Revisited

• Collapsing to Knots and Links

• Stability in Processive DNA Recombination

• Afterthoughts - Farey Series, Continued Fractions, Pick's Theorem and Ford Circles
  • Farey Series and Continued Fractions
  • Pick's Theorem
  • Ford Circles

• References

The following sections are included:

• Introduction

• Bracket Polynomial and Jones Polynomial

• Khovanov Homology and the Cube Category

• Khovanov Homology

• The Cube Category and the Tangle Cobordism Structure of Khovanov Homology

• Other Frobenius Algebras and Rasmussen's Theorem

• The Simplicial Structure of Khovanov Homology

• Quantum Comments

• Discussion

• References

In this survey paper we present the L–moves between braids and how they can adapt and serve for establishing and proving braid equivalence theorems for various diagrammatic settings, such as for classical knots, for knots in knot complements, in c.c.o. 3–manifolds and in handlebodies, as well as for virtual knots, for flat virtuals, for welded knots and for singular knots. The L–moves are local and they provide a uniform ground for formulating and proving braid equivalence theorems for any diagrammatic setting where the notion of braid and diagrammatic isotopy is defined, the statements being first geometric and then algebraic.

We consider knot theories possessing parity: each crossing is decreed odd or even according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, then this leads to a possibility of constructing new invariants and proving minimality and non-triviality theorems for knots from these classes, and constructing maps from knots to knots.

Our main example is virtual knot theory and its simplification, free knot theory. By using Gauss diagrams, we show the existence of non-trivial free knots (counterexample to Turaev's conjecture) and construct simple yet deep invariants that rely on parity. Some invariants are valued in graph-like objects and some other are valued in groups.

We discuss applications of parity to virtual knots and ways of extending well-known invariants.

The existence of a non-trivial parity for classical knots remains an open problem.

This paper contains selected topics from four lectures given at the Abdus Salam International Centre for Theoretical Physics in May 2009. We introduce the study of the influence of knotting and linking on the spatial characteristics of linear and ring polymer chains with examples of scientific interest. We describe a few basic concepts of the geometry and topology of knots and measures of the spatial shape of open and closed polymer chains. We then present some fundamental mathematical results concerning them. Next we discuss random sampling methods of collections of open and closed chains that are employed to provide estimates of the spatial properties of the chains. Finally, we discuss implementations of the sampling algorithms, survey consequences of theoretical and experimental results, and discuss some interesting problems deserving further research.

The following sections are included:

• Introduction
  • Setting the scene
  • Background
  • Knot diagrams and moves
  • Links and linking number
  • Framed links
  • Satellites

• Homfly invariants

• General Homfly skein theory
  • Composition
  • Closure

• The skein of the annulus
  • Meridian maps
  • Partitions
  • Branching rules in C

• Symmetric functions and the skein of the annulus
  • Symmetric functions
  • Construction of the basis elements

• Unitary quantum invariants
  • Basic constructions of quantum invariants

• Manifold invariants
  • Surgery presentation
  • Manifolds with boundary
  • Evaluation of knot invariants
  • Level invariants for manifolds

• References

This paper is base on talks which I gave in May, 2010 at Workshop in Trieste (ICTP). In the first part we present an introduction to knots and knot theory from an historical perspective, starting from Summerian knots and ending on Fox 3-coloring. We show also a relation between 3-colorings and the Jones polynomial. In the second part we develop the general theory of Fox colorings and show how to associate a symplectic structure to a tangle boundary so that tangles becomes Lagrangians (a proof of this result has not been published before). We also discuss rational moves on links and their relation to Fox colorings.

In this brief presentation, we would like to present our attempts of detecting chirality and mutations from Chern-Simons gauge theory. The results show that the generalised knot invariants, obtained from Chern-Simons gauge theory, are more powerful than Jones, HOMFLYPT and Kauffman polynomials. However the classification problem of knots and links is still an open challenging problem.

We present some commonly-used models for polymers in different experimental conditions, including some discussion of implementation issues. We then analyze how the size and shape of polymers are affected by length and the existence of knotting.

In these lectures we present for the first time a mathematical reconstruction of what might have been Gauss' own derivation of the linking number of 1833, and we provide also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and intersection number. The reconstruction offered here is entirely based on an accurate study of Gauss' own work on terrestrial magnetism and it is complemented by the independent analysis and discussion made by Maxwell in 1867. Since the linking number interpretations in terms of degree, signed crossings and intersection index play such an important rôle in modern mathematical physics, we offer a direct proof of their equivalence, and we provide some examples of linking number computation based on oriented area information. The material presented in these lectures forms an integral part of a paper that will appear in the Journal of Knot Theory and Its Ramifications.

This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial.

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