Introduction to Knot Theory Spring 2019

Meike Akveld
Tommaso Goldhirsch

Mon 15-17, HG D1.1
Exercise Class
Thu 17-18, HG D7.1
Definition of a knot and of equivalent knots. Definition of a knot invariant and some elementary examples. Various operations on knots. Knot polynomials (Jones, ev. Alexander.....).

Tentative list of topics to be covered:

  1. Historical background
  2. Informal introduction and outlook
  3. Official introduction
  4. Simple knot invariants
  5. The Jones polynomial
  6. Alternating knots
  7. Surfaces (an overview)
  8. Seifert surfaces
  9. The Alexander polynomial

Lecture Summaries

Week Chapter(s) Topics Files and Slides
01 Chapter 0 and 1 History of knot theory, informal definition of a knot and knot equivalence, crossing number, unknot, operations on knots (mirror image, reflection, connected sum), alternating knots, the unknotting number
02 Chapter 2 Definition of a knot, of knot equivalence, wild and tame knots, polygonal knots, Delta moves, regular projections and knot diagrams A wild knot and a polygonal knot
03 Chapter 2 and 3 Regular projections are generic, a strange upperbound, crossing number, unknotting number, definition of links, linking number A complicated unknot and a paper by Coward and Lackenby (2010)
04 Chapter 3 3-colourings and p-colourings Example of p-colourings for two simple knots
05 Chapter 4 Definition of the Jones polynomial using the bracket polynomial. Definition of states of diagrams. The original paper of V.R.Jones
A nontrivial link with trivial Jones polynomial
Two different knots with the same Jones Polynomial
06 Chapter 4 Description of Kauffman bracket in terms of states, axiomatic definition of the Jones polynomial and its uniqueness using the Skein relation and the concept of complexity, mirror images revisited A table of the Jones Polynomial from Lickorish
Different mirror images with the same Jones Polynomial
07 Chapter 5 Alternating knots: Definition of reduced and minimal knot diagrams, Theorems about relation of breath of Kauffman bracket (a knot invariant) and the number of crossings, statement of the first Tait Conjecture. Spot the non-alternating knot diagrams
Example how the states relates to the number of closed loops
08 No lecture Sechseläuten
09 Chapter 5 Alternating knots: Proof of the first Tait conjecture, definition of the HOMFLY and the Alexander-Conway polynomial and some properties of the latter. Properties of the Conway polynomial
Proof of proposition from Lickorish' book
10 Chapter 6 A crash course on surfaces: Repetition of compact or closed n-manifolds, quotient topology, adding handles, cross handles and crosscaps. Definition of combinatorial surfaces. Curves and arcs on surfaces (1-sides, 2-sides, separating, non-separating) and non-orientable surfaces. Which surface is this?
11 Chapter 6 Euler characteristic as an invariant of surfaces, surfaces represented by polygons with a gluing pattern. Genus of a surface. Classification theorems for closed surfaces and for surfaces with boundary. Classification of surfaces
12 Chapter 7 Seifert surfaces: Definition of a Seifert surface, proof of existence, genus of a knot, Theorem about additivity of the genus, prima knots. Construction of a Seifert surface
13 Chapter 8 Seifert matrices: Definition, Seifert- (or S-) equivalence of matrices, Theorem about equivalent knots and their Seifert matrices Example for calculating a Seifert matrix
Nested circles
See Chapter 5 of "Knot Theory & Its Applications" by Kunio Murasugi
14 Chapter 8 The Alexander polynomial: The determinant of a knot, the Alexander polynomial and some of its properties, skein relation, some examples Some examples of knots with different and equal Jones and Alexander polynomials
See Chapter 6 of "Knot Theory & Its Applications" by Kunio Murasugi

The new exercises will be posted here on Monday. We expect you to look at the problems and prepare questions for the exercise class on Thursday.

Please hand in your solutions by the following Monday in HG J68. Your solutions will usually be corrected and returned in the following exercise class or, if not collected, returned to the box in HG J68.

exercise sheet due by solutions
Exercise sheet 1: Informal introduction and outlook Monday 25.02.2019 Solutions 1
Exercise sheet 2: Formal introduction to knots Monday 4.03.2019 Solutions 2
Exercise sheet 3: Reidemeister moves and some simple knot invariants Monday 11.03.2019 Solutions 3
Exercise sheet 4: p-colourings Monday 18.03.2019 Solutions 4
Exercise sheet 5: the Jones polynomial I Monday 25.03.2019 Solutions 5
Exercise sheet 6: the Jones polynomial II Monday 01.04.2019 Solutions 6
Exercise sheet 7: Alternating knots Monday 15.04.2019 Solutions 7
Exercise sheet 8: More about polynomial invariants Monday 29.04.2019 Solutions 8
Exercise sheet 9: Crashcourse on surfaces I Monday 06.05.2019 Solutions 9
Exercise sheet 10: Crashcourse on surfaces II Monday 13.05.2019 Solutions 10
Exercise sheet 11: Seifert surfaces Monday 20.05.2019 Solutions 11
Exercise sheet 12: Seifert matrices Monday 27.05.2019 Solutions 12
Exercise sheet 13: the Alexander polynomial Monday 03.06.2019 Solutions 13

There will be NO exercise class on the 11th of April. We will have an exercise class during the last week of the semester; more informations will follow.

Thu 17-18HG D 7.1 Tommaso Goldhirsch en