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The course will be delivered in English. 

Low-dimensional topology

Programma

The topics will include a selection from the following themes:

 

- Introduction to knot theory.  Diagrams.  Framed knots.  (6 hours)

- Dehn surgery and filling.  Handles.  4D interpretation.  Dehn twists. 

  Existence of a surgery presentation.  Kirby moves.  (4 hours)

- Wirtinger presentation and p-colorings. Tietze theorem. (3 hours)

- Morse functions and 3D smooth Jordan-Schoenflies theorem.

  Tori in the 3-sphere. (2 hours)

- Connected sum of knots.  Genus.  Additivity.  Prime decomposition.  (4 hours)

- The braid group.  Alexander and Markov theorems.  (3 hours)

- The loop theorem and applications to knot theory.  (4 hours)

- Ideal triangulations and special spines.  Normal surfaces.  Connected sum

  of 3-manifolds.  Existence and uniqueness of the prime decomposition of a

  3-manifold.  (8 hours)

- Kauffman bracket and polynomial, Jones polynomial. 

  Tait conjectures.  (4 hours)

Prerequisiti

prerequisites are a basic mathematical literacy, general topology, the fundamental group and integer homology, but the lectures will be essentially self-contained. 

Svolgimento

 Il corso si svolgerà dal 6 marzo al 18 maggio presso le Aule di Palazzo del Broletto Piazza della Vittoria, 15 - Pavia

Calendario:

In via di definizione

Bibliografia

Useful books:

 

Burde-Zieschang-Heusener - Knots

 

Lickorish – An introduction to knot theory

 

Rolfsen – Knots and links

 

Sossinsky-Prasolov – Knots, links, braids and 3-manifolds

 

Jaco – Lectures on three-manifold topology

 

Matveev-Fomenko – Algorithmic and computer methods in three-dim-ensional topology

 

Matveev - Algorithmic topology and classification of 3-manifolds 

 

Carlo Petronio

Professore ordinario di Geometria presso il Dipartimento di Matematica dell' Università di Pisa

Classe : Scienze Tecnologie e Società

Ambito : Scienze e Tecnologie

Periodo: Semestre II

Anno accademico: 2019-2020

Luogo : Pavia

Durata : 25 ore