Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors). ETSU does not have a formal class on knot theory, but these notes may be used in an Independent Study (MATH 4900) or as a supplement to Introduction to Topology (MATH 4357/5357).

These notes are based on Charles Livingston's Knot Theory book. The book is a bit informal and, in these notes, some claims are not proved but instead justified with statements like "Livingston claims..." Such claims are sometimes justified by referencing sources which provide detailed proofs. There are also some typographical errors in the Livingston book and the known errors are corrected in these notes.

Notes are in preparation for a more rigorous class in knot theory. See my online page for Graduate Knot Theory Class Notes based on W. B. Raymond Lickorish's An Introduction Knot Theory, Graduate Texts in Mathematics 175, (NY: Springer-Verlag, 1997).

• Chapter 1. A Century of Knot Theory.
• Chapter 2. What is a Knot?
• Chapter 3. Combinatorial Techniques.
• Chapter 4. Geometric Techniques.
• Chapter 5. Algebraic Techniques.
• Chapter 6. Geometry, Algebra, and the Alexander Polynomial.
• Chapter 7. Numerical Invariants.
• Chapter 8. Symmetries of Knots.
• Chapter 9. High-Dimensional Knot Theory.
• Chapter 10. New Combinatorial Techniques.
• Appendix 1. Knot Table.
• Appendix 2. Alexander Polynomials.

Chapter 1. A Century of Knot Theory.

• Chapter 1. A Century of Knot Theory. Chapter 1 notes
• Study Guide 1

Chapter 2. What is a Knot?

• Section 2.1. Wild Knots and Unknottings. Section 2.1 notes
• Section 2.2. The Definition of a Knot. Section 2.2 notes
• Section 2.3. Equivalence of Knots, Deformations. Section 2.3 notes
• Section 2.4. Diagrams and Projections. Section 2.4 notes
• Section 2.5. Orientations. Section 2.5 notes
• Study Guide 2

Chapter 3. Combinatorial Techniques.

• Section 3.1. Reidemeister Moves. Section 3.1 notes
• Section 3.2. Colorings. Section 3.2 notes
• Supplement. Proofs from Section 3.2. Proofs of Theorems in Section 3.2
• Supplement. Printouts of Proofs from Section 3.2. Print file of Proofs of Theorems in Section 3.2
• Section 3.3. A Generalization of Colorability, mod p Labelings. Section 3.3 notes
• Section 3.4. Matrices, Labelings, and Determinants. Section 3.4 notes
• Section 3.5. The Alexander Polynomial. Section 3.5 notes
• Study Guide 3

Chapter 4. Geometric Techniques.

• Chapter 4. Geometric Techniques. Chapter 4 notes
• Section 4.1. Surfaces and Homeomorphisms. Section 4.1 notes
• Section 4.2. The Classification of Surfaces. Section 4.2 notes
• Supplement. Proofs from Section 4.2. Proofs of Theorems in Section 4.2
• Supplement. Printouts of Proofs from Section 4.2. Print file of Proofs of Theorems in Section 4.2
• Section 4.3. Seifert Surfaces and the Genus of a Knot. Section 4.3 notes
• Supplement. Proofs from Section 4.3. Proofs of Theorems in Section 4.3
• Supplement. Printouts of Proofs from Section 4.3. Print file of Proofs of Theorems in Section 4.3
• Section 4.4. Surgery on Surfaces. Section 4.4 notes
• Supplement. Proofs from Section 4.4. Proofs of Theorems in Section 4.4
• Supplement. Printouts of Proofs from Section 4.4. Print file of Proofs of Theorems in Section 4.4
• Section 4.5. Connected Sums on Knots and Prime Decompositions. Section 4.5 notes
• Supplement. Proofs from Section 4.5. Proofs of Theorems in Section 4.5
• Supplement. Printouts of Proofs from Section 4.5. Print file of Proofs of Theorems in Section 4.5
• Study Guide 4

Chapter 5. Algebraic Techniques.

• Section 5.1. Symmetric Groups. Section 5.1 notes
• Section 5.2. Knots and Groups. Section 5.2 notes
• Supplement. Proofs from Section 5.2. Proofs of Theorems in Section 5.2
• Supplement. Printouts of Proofs from Section 5.2. Print file of Proofs of Theorems in Section 5.2
• Section 5.3. Conjugation and the Labeling Theorem. Section 5.3 notes
• Supplement. Proofs from Section 5.3. Proofs of Theorems in Section 5.3
• Supplement. Printouts of Proofs from Section 5.3. Print file of Proofs of Theorems in Section 5.3
• Section 5.4. Equations in Groups and the Group of a Knot. Section 5.4 notes
• Supplement. Proofs from Section 5.4. Proofs of Theorems in Section 5.4
• Supplement. Printouts of Proofs from Section 5.4. Print file of Proofs of Theorems in Section 5.4
• Section 5.5. The Fundamental Group. Section 5.5 notes
• Study Guide 5

Chapter 6. Geometry, Algebra, and the Alexander Polynomial.

• Section 6.1. The Seifert Matrix. Section 6.1 notes
• Section 6.2. Seifert Matrices and the Alexander Polynomial. Section 6.2 notes
• Supplement. Proofs from Section 6.2. Proofs of Theorems in Section 6.2
• Supplement. Printouts of Proofs from Section 6.2. Print file of Proofs of Theorems in Section 6.2
• Section 6.3. The Signature of a Knot, and the other S-Equivalence Invariants. Section 6.3 notes
• Supplement. Proofs from Section 6.3. Proofs of Theorems in Section 6.3
• Supplement. Printouts of Proofs from Section 6.3. Print file of Proofs of Theorems in Section 6.3
• Section 6.4. Knot Groups and the Alexander Polynomial. Section 6.4 notes
• Study Guide 6

Chapter 7. Numerical Invariants.

• Section 7.1. Summary of Numerical Invariants. Section 7.1 notes
• Section 7.2. New Invariants. Section 7.2 notes
• Section 7.3. Braids and Bridges. Section 7.3 notes
• Section 7.4. Relations Between the Numerical Invariants. Section 7.4 notes
• Section 7.5. Independence of Numerical Invariants. Section 7.5 notes
• Supplement. Proofs from Section 7.5. Proofs of Theorems in Section 7.5
• Supplement. Printouts of Proofs from Section 7.5. Print file of Proofs of Theorems in Section 7.5
• Study Guide 7

Chapter 8. Symmetries of Knots.

• Chapter 8. Symmetries of Knots. Chapter 8 notes
• Section 8.1. Amphicheiral and Reversible Knots. Section 8.1 notes
• Section 8.2. Periodic Knots. Section 8.2 notes
• Supplement. Proofs from Section 8.2. Proofs of Theorems in Section 8.2
• Supplement. Printouts of Proofs from Section 8.2. Print file of Proofs of Theorems in Section 8.2
• Section 8.3. The Murasugi Conditions. Section 8.3 notes
• Section 8.4. Periodic Seifert Surfaces and Edmonds' Theorem. Section 8.4 notes
• Supplement. Proofs from Section 8.4. Proofs of Theorems in Section 8.4
• Supplement. Printouts of Proofs from Section 8.4. Print file of Proofs of Theorems in Section 8.4
• Section 8.5. Applications of the Murasugi and Edmonds Conditions. Section 8.5 notes
• Supplement. Proofs from Section 8.5. Proofs of Theorems in Section 8.5
• Supplement. Printouts of Proofs from Section 8.5. Print file of Proofs of Theorems in Section 8.5
• Study Guide 8

Chapter 9. High-Dimensional Knot Theory.

• Chapter 9. High-Dimensional Knot Theory. Chapter 9
• Section 9.1. Defining High-Dimensional Knots. Section 9.1 notes
• Section 9.2. Three Dimensions from a 2-Dimensional Perspective. Section 9.2 notes
• Section 9.3. Three-Dimensional Cross-Sections of a 4-Dimensional Knot. Section 9.3 notes
• Section 9.4. Slice Knots. Section 9.4 note
• Supplement. Proofs from Section 9.4. Proofs of Theorems in Section 9.4
• Supplement. Printouts of Proofs from Section 9.4. Print file of Proofs of Theorems in Section 9.4
• Section 9.5. The Knot Concordance Group. Section 9.5 notes
• Supplement. Proofs from Section 9.5. Proofs of Theorems in Section 9.5
• Supplement. Printouts of Proofs from Section 9.5. Print file of Proofs of Theorems in Section 9.5
• Study Guide 9

Chapter 10. New Combinatorial Techniques.

• Chapter 10. New Combinatorial Techniques. Chapter 10 notes
• Section 10.1. The Conway Polynomial of a Knot. Section 10.1 notes
• Section 10.2. New Polynomial Invariants. Section 10.2 notes
• Section 10.3. Kaufman's Bracket Polynomial. Section 10.3 notes
• Study Guide 10

Appendix 1. Knot Table. The Rolfsen Knot Table on The Knot Atlas page includes images of all knots with 10 or fewer crossings (accessed 2/25/2021). Livingston's book has images of all knots with 9 or fewer crossings in its Appendix 1.

Appendix 2. Alexander Polynomials. Alexander Stoimenow has an online Table of Alexander Polynomials for all knots with 10 or fewer crossings (the notation is explained here; accessed 2/25/2021).

Return to Bob Gardner's home page