## Introductory Lectures on Knot Theory: Selected Lectures Presented at the Advanced School and Conference on Knot Theory and Its Applications to Physics and Biology, ICTP, Trieste, Italy, 11 - 29 May 2009This volume consists primarily of survey papers that evolved from the lectures given in the school portion of the meeting and selected papers from the conference.Knot theory is a very special topological subject: the classification of embeddings of a circle or collection of circles into three-dimensional space. This is a classical topological problem and a special case of the general placement problem: Understanding the embeddings of a space X in another space Y. There have been exciting new developments in the area of knot theory and 3-manifold topology in the last 25 years. From the Jones, Homflypt and Kauffman polynomials, quantum invariants of 3-manifolds, through, Vassiliev invariants, topological quantum field theories, to relations with gauge theory type invariants in 4-dimensional topology.More recently, Khovanov introduced link homology as a generalization of the Jones polynomial to homology of chain complexes and Ozsvath and Szabo developed Heegaard-Floer homology, that lifts the Alexander polynomial. These two significantly different theories are closely related and the dependencies are the object of intensive study. These ideas mark the beginning of a new era in knot theory that includes relationships with four-dimensional problems and the creation of new forms of algebraic topology relevant to knot theory. The theory of skein modules is an older development also having its roots in Jones discovery. Another significant and related development is the theory of virtual knots originated independently by Kauffman and by Goussarov Polyak and Viro in the '90s. All these topics and their relationships are the subject of the survey papers in this book.It is a remarkable fact that knot theory, while very special in its problematic form, involves ideas and techniques that involve and inform much of mathematics and theoretical physics. The subject has significant applications and relations with biology, physics, combinatorics, algebra and the theory of computation. The summer school on which this book is based contained excellent lectures on the many aspects of applications of knot theory. This book gives an in-depth survey of the state of the art of present day knot theory and its applications. |

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### Contents

On the Unification of Quantum 3Manifold Invariants A Beliakova and T Le | 1 |

A Survey of Quandle Ideas J Scott Carter | 22 |

Combinatorics of Vassiliev Invariants S Chmutov | 54 |

Braid Order Sets and Knots P Dehornoy | 77 |

Finding Knot Invariants from Diagram Colouring R Fenn | 97 |

Exceptional Dehn Filling C McA Gordon | 124 |

GraphLinks D P Ilyutko and V O Manturov | 135 |

Diagrammatic Knot Properties and Invariants S V Jablan and R Sazdanovic | 162 |

Free Knots and Parity V O Manturov | 321 |

An Introduction to the Study of the Inuence of Knotting on the Spatial Characteristics of Polymers K C Millett | 346 |

Knots Satellites and Quantum Groups H R Morton | 379 |

The Trieste Look at Knot Theory J H Przytycki | 407 |

Dectection of Chirality and Mutations of Knots and Links R Pichai | 442 |

The Study of Sizes and Shapes of Polymers E J Rawdon | 457 |

Derivation and Interpretation of the Gauss Linking Number R L Ricca and B Nipoti | 482 |

Introduction to Virtual Knot Theory L H Kau man | 502 |

Hard Unknots and Collapsing Tangles L H Kauffman and S Lambropoulou | 187 |

Khovanov Homology L H Kauffman | 248 |

Braid Equivalences and the Lmoves S Lambropoulou | 281 |

List of Participants | 543 |

Programme | 547 |

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### Common terms and phrases

3-manifolds 4-graph algebraic arrow polynomial biquandle boundary bracket polynomial braid equivalence braid group categorification chain chord diagram classical knots closure cocycle colors components compute continued fraction corresponding defined Definition denote edges element ellipsoid example Figure finite flat virtual formula framed graph free knots Gauss diagram geometric graph-link hard unknot integral isotopy Jones polynomial Kauffman Khovanov homology knot diagram knot invariants knot or link knot theory knot type knots and links L–move labeled Lemma link diagram linking number looped interlacement graphs manifolds Math Mathematics minimal diagrams mixed braid module non-trivial obtained oriented pair parity polygons polymers proof Przytycki quandle quantum rack rational knots rational tangles Reidemeister moves relation result skein space strands Theorem Topology trefoil unknot unknotting number Vassiliev invariants vectors vertex vertices virtual crossings virtual diagram virtual knot theory virtual knots virtual links