| Date |
Lecture Content |
| 8/19/25 |
Introduction. Dehn Surgery on knots. |
Saveliev, Lectures on the Topology of 3-Manifolds: Lecture 2 §2.1-2.3 (see also Lecture 1 for some relevant background)
Further reading: Gordon, Dehn surgery and 3-Manifolds (2006). Many examples and exercises. (Not up to date, though.)
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| 8/21/25 |
Wirtinger presentations. Fundamental groups of surgeries. |
|
Rolfsen, Knots and Links: The Wirtinger Presentation §3.D
|
| 8/26/25 |
SU(2) and quaternions. Representations spaces and conjugation action. |
Saveliev, Lectures on the Topology of 3-Manifolds: Lecture 13
house notes: SU(2) as unit quaternions
|
| 8/28/25 |
Stabilizer types of representations. The reducible part of the character variety. |
| house notes: Stablizers and orbits, and the reducible characters
|
| 9/2/25 |
SU(2) character varieties of knots. Example computations. The pillowcase. |
|
Kirk, SU(2) Representation Varieties of 3-manifolds, Gauge Theory Invariants, and Surgery on Knots: pdf Ch.1
|
| 9/4/25 |
Torus knots. Herald's Theorem on the structure of the SU(2) character variety (intro). |
|
Klassen, Representations of knot groups in SU(2): pdf IA, IB
|
| 9/9/25 |
Perturbations, Herald's Theorem. Character varieties, amalgamated groups, and Dehn surgery. |
Herald, Legendrian cobordism and Chern-Simons theory on 3-manifolds: link
Herald, Kirk, Holonomy perturbations in a cylinder... link, see especially §2
|
| 9/11/25 |
Understanding Dehn surgery in the pillowcase. Proving first cases of Thm 1 from from [KM] |
|
Kirk paper above, KM paper above
|
| 9/16/25 |
Proof of Thm 1 from [KM]. Perturbations in a thickened torus. |
|
KM paper above; Herald-Kirk paper above does more involved perturbations
|
| 9/18/25 |
Local structure of representation spaces. Zariski tangent spaces. Group cohomology. |
Saveliev, Lectures on the Topology of 3-Manifolds: Lecture 15
Weil, Remarks on the cohomology of groups: pdf
|
| 9/23/25 |
Homology with local coefficients. Zariski tangent space computations for the pillowcase. |
|
Kirk paper above, Hatcher Algebraic Topology § 3.H
|
| 9/25/25 |
Local structure at a reducible, and relation to the Alexander polynomial. |
Lickorish, An introduction to Knot Theory Ch 6
Milnor, Infinite cyclic coverings §1-2
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| 9/30/25 |
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| 10/2/25 |
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