Knots and 3-manifolds

Math 87100, Spring 2022

Thursdays, 11:45 - 1:45 pm, Room 6417

Mathematics Program, The Graduate Center, CUNY

Instructor: Abhijit Champanerkar

Office: 4307

Email :

Office Hours: Thursdays 1:45 - 2:45 pm

Course description This course will cover different aspects of knot theory and 3-manifold topology and the deep interactions between the two fields. We will cover the following topics: 3-manifold topology and geometry including Dehn surgery, Heegaard Spittings, Seifert Fibered Spaces, Prime Decomposition, JSJ decomposition, Eight 3-dimensional geometries, hyperbolic 3-manifolds. Knot diagrams, Reidemeister moves, diagrammatic invariants, popular knot families, Braids, Alexander and Markov theorems, Seifert surfaces and classical invariants, Alexander and Jones polynomials, Quantum invariants of knots, Hyperbolic knots and geometric invariants, Representation of knots groups, leading conjectures in knots and 3-manifolds.

Reference Books

• Martelli: An introduction to Geometric Topology by Bruno Martelli, available online.
• Hatcher: Notes on basic 3-manifold topology by Allen Hatcher, available online.
• Rolfsen: Knots and Links by Dale Rolfsen, Publish or Perish
• Purcell: Hyperbolic Knot Theory by Jessica Purcell online
• Cromwell: Knots and links by Peter Cromwell
• Lickorish: An introduction to knot theory by W.B.Raymond Lickorish
• Prasolov-Sossinsky: Knots, links, braids and 3-manifolds by V.V.Prasolov and A.B.Sossinsky

I am covering topics from a combination of sources including books and notes above. The topic-wise references given below is a good approximation of the topics we are covering in class.

Date	Topic	References/Reading	Suggested HW
Feb 3rd	3M1 Preliminaries & 3-manifolds	Triangulations of manifolds   Triangulation of surfaces Classification of surfaces: Conway's ZIP proof Zeeman's proof Morse theory and surfaces	HW 1
Feb 10th	3M2 Dehn surgery, Heegaard Splittings and Lickorish-Wallace Theorem	Rolfsen: Chpt 9F, 9I, Lickorish: Chpt 12, Lens Spaces	HW 2
Feb 17th	3M3 Alexander's Theorem on irreducibility of 3-space, Normal surfaces	A's Theorem - Hatcher: pg 1-4, Normal surfaces - Martelli: 9.2.8-9.2.9	HW3
Feb 24th	3M4 Kneser-Milnor Prime decomposition of 3-manifolds, Seifert Fibered spaces	Prime decomposition - Hatcher: pgs 5-11, Martelli: 9.2.10 SFS - Martelli: 10.3.1-10.3.4	HW4
Mar 3rd	3M5 Seifert Fibered Spaces, JSJ decomposition	SFS - Martelli: 10.4.2, 10.4.3 JSJ Martelli: 11.5, Hatcher: pg 13 - 29	HW5
Mar 10th	KT1 Knots, links, Reidemeister moves and fundamental group	Prasolov-Sossinsky: Chpt 1 Rolfsen: 3A-3F
Mar 17th	KT2 Dehn-Papakyriakopoulous Lemma, Knot groups, Braids	Rolfsen: Chpt 4 Lickorish: Chpt 11 Prasolov-Sossinsky: Chpt 3
Mar 24th	KT3 Seifert surfaces	Cromwell: Chpt 5 Lickorish: Chpt 2
Mar 31st	KT4 Seifert matrix, Alexander polynomial and properties	Cromwell: 6.5 - 6.8, 7.1 - 7.4 Lickorish: Chpt 6
Apr 7th	KT5 Jones polynomial, properties and Tait Conjecture	Lickorish: Chpt 3, 5 Cromwell: Chpt 9
Apr 14th	KT6 Alternating knots - diagrammatics and topology	Lickorish: Chpts 4, 5
Apr 21st	No class - Spring break
Apr 28th	G3M1 Geometrization of 3-manifolds	References for Geometry of 3-manifolds Geometries of 3-manifolds by Peter Scott, Bulletin of LMS, 15 (1983) online. Geometric structures on 3-manifolds by Francis Bonahon, Handbook of Geometric Topology, available online.
May 5th	G3M2 Hyperbolic 3-manifolds 1	References for Geometry of 3-manifolds
May 12th	G3M3 Hyperbolic 3-manifolds 2	References for Geometry of 3-manifolds   SnapPy   KnotAtlas   CompuTop
	Final Report - Review of paper	List of papers

Related Seminars

• CUNY Geometry and Topology Seminar, Tuesdays 4:15 - 5:15 pm, GC
• Columbia Geometric Topology Seminar, Fridays 2 - 3 pm, Columbia University
• Geometry and Topology Student Seminar, Tuesdays 1:30 - 3 pm, GC

Additional references

• An introduction to 3-manifolds by Jennifer Schultens, AMS-GSM series.
• An introduction to 3-manifolds by Stephan Friedl, available online.
• Notes on basic 3-manifold topology by Allen Hatcher, available online.
• Geometries of 3-manifolds by Peter Scott, Bulletin of LMS, 15 (1983) online.
• Geometric structures on 3-manifolds by Francis Bonahon, Handbook of Geometric Topology, available online.

Background image: London Tsai
John Baez on Hopf fibration