An Introduction to Math Beyond Calculus: Knot Theory

The following is a **tentative** syllabus for the course.

Week | Lectures | Sections in Text | Brief Description | |
---|---|---|---|---|

1 | 3/30 | Ch. 1 Sec, 2.1 - 2.2 |
Introduction, history, and definitions of knots | |

4/1 | 2.3 - 3.1 | Equivalence of knots, Reidemeister moves | ||

4/3 | 3.2 - 3.3 | Colorings of knots | ||

2 | 4/6 | 3.3 - 3.4 | Generalizations of colorability | |

4/8 | 3.5 | The Alexander polynomial | ||

4/10 | 4.1 | Surfaces | ||

3 | 4/13 | 4.2 | Classification of surfaces | |

4/15 | 4.3 | Seifert surfaces and knot genus | ||

4/17 | 4.4 - 4.5 | Connected sums and prime decomposition, relations to invariants |
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4 | 4/20 | 5.1 | Definition of a group, symmetric groups | |

4/22 | 5.2 - 5.3 | Labelling knots with symmetric groups | ||

4/24 | 5.4 | The group of a knot | ||

5 | 4/27 | 6.1 - 6.2 | Seifert matrices | |

4/29 | 6.2 | The Alexander polynomial, part 2 | ||

5/1 |
Midterm Exam |
Material covered from Chapters 1-5. | ||

6 | 5/4 | 6.3, 7.1 | The knot signature and other knot invariants | |

5/6 | 7.2 - 7.3 | New invariants, braids, and bridges | ||

5/8 | 7.4 - 7.5 | Relationships and independence of invariants | ||

7 | 5/11 | 10.1 | The Conway polynomial | |

5/13 | 10.2 | The Bracket polynomial | ||

5/15 | 9.1 | Higher-dimensional knots | ||

8 | 5/18 | 9.2 | Three dimensions from a 2-dimensional perspective | |

5/20 | 9.3 | Four dimensions from a 3-dimensional perspective | ||

5/22 | 9.4 | Slice knots | ||

9 | 5/25 | Memorial Day: no classes | ||

5/27 | 9.4 - 9.5 | The knot concordance group, part 1 | ||

5/29 | 9.5 | The knot concordance group, part 2 | ||

10 | 6/1 | |||

6/3 |