Math 607 Winter 2020

Instructor: Robert Lipshitz

Basic course information

Time: 12:00-12:50 MWF
Place: Fenton 119
Office hours: by appointment.
Final exam: none.

Prerequisites

Some basic algebraic topology (homology, fundamental group, covering spaces) and some basic knowledge of smooth manifolds (Math 532 suffices). Students outside the mathematics Ph.D. program must obtain the instructor's permission to enroll in this course.

Description and goals

This course is an introduction to Khovanov homology of knots and the Heegaard Floer homology of knots and 3-manifolds, with an emphasis on their topological applications. One goal is to discuss a range of motivating ideas, recent results, and open questions in this area.

Course requirements and policies

Students are required to turn in solutions to two homework problems during the quarter. At least one of these must be turned in by mid-quarter.

Students with disabilities

I, and the University of Oregon in general, are committed to an inclusive learning environment. If you have a disability which may impact your performance on exams, please contact the Accessible Education Center to discuss appropriate accommodations. If there are other disability-related barriers to your participation in the course, please either discuss them with me directly or consult with the Accessible Education Center.

Schedule

Week Date Topic
1 1/6

Knot theory basics

Knots, isotopy, ambient isotopy. Knot diagrams. Knot genus.

  1/8

Prime knots, unique factorization. Hyperbolic knots, satellite knots, torus knots.

  1/10

Slice genus, slice knots, ribbon knots, concordance groups.

The last day to drop the class is Saturday.

2 1/13

Alexander and Jones polynomials

Alexander polynomial definition, genus bound.

  1/15

Skein relation for Alexander polynomial. Jones polynomial, HOMFLY-PT polynomial.

  1/17

Kauffman states and the Jones polynomial. Statement of Tait conjectures.

3 1/20

Martin Luther King Day (no class)

  1/22

Khovanov homology: definition, example computation, Euler characteristic.

  1/24

Invariance of Khovanov homology

4 1/27

Khovanov homology for tangles and cobordisms

Khovanov's arc algebras and bimodules

  1/29

Bar-Natan's picture world

  1/31

Tangle cobordisms

5 2/3

Lee spectral sequence and the s-invariant

Deformations of Khovanov homology, equivalence of deformations, deformations which diagonalize

  2/5

Rasmussen's s-invariant

  2/7

Freedman-Quinn and exotic smooth structures on R^4

6 2/10

Khovanov stable homotopy type

  2/12

 

  2/14

 

7 2/17

Grid diagrams and knot Floer homology

Definition of knot Floer homology via grid diagrams. tau invariant.

  2/19

Aspects of the combinatorial proof of invariance

  2/21

Legendrian and transverse knots, invariants of them in knot Floer and Khovanov homology

8 2/24

Heegaard diagrams and holomorphic disks

  2/26

 

  2/28

 

9 3/2

Surgery formulas and applications

  3/4

 

  3/6

 

10 3/9

The Ozsváth-Szabó and Kronheimer-Mrowka spectral sequences and some detection results

  3/11

 

  3/13

 

 

References

References for details and further reading.

  • [BJ] Bröcker, Th.; Jänich, K. Introduction to differential topology.
  • [Bl] Blanchet, Christian. "An oriented model for Khovanov homology." MR2647055.
  • [BN1] Bar-Natan, Dror. "On Khovanov's categorification of the Jones polynomial." MR1917056.
  • [BN2] Bar-Natan, Dror. "Khovanov's homology for tangles and cobordisms." MR2174270.
  • [BP] Borodzik, Maciej; Politarczyk, Wojciech. "Khovanov homology and periodic links." arXiv:1704.07316.
  • [BPS] Borodzik, Maciej; Politarczyk, Wojciech; Silvero, Marithania. "Khovanov homotopy type, periodic links and localizations." arXiv:1807.08795.
  • [Ca] Caprau, Carmen. "sl(2) tangle homology with a parameter and singular cobordisms." MR2443094.
  • [CD] Cochran, Tim; Davis, Christopher. "Counterexamples to Kauffman's conjectures on slice knots." MR3318151.
  • [Ch] Chbili, Nafaa. "Equivalent Khovanov homology associated with symmetric links." MR2779238.
  • [CMW] Clark, David; Morrison, Scott; Walker, Kevin. "Fixing the functoriality of Khovanov homology." MR2496052.
  • [Cor] Cornish, James. "Sutured annular Khovanov homology and two periodic braids." arXiv:1606.03034.
  • [Cr] Cromwell, Peter. Knots and Links.
  • [Et] Etnyre, John. "Legendrian and transversal knots." MR2179261. Available here.
  • [Fr] Friedl, Stefan. "An introduction to 3-manifolds and their fundamental groups." Available here.
  • [Gr1] Greeene, Joshua. "The lens space realization problem." MR3010805.
  • [Gr2] Greeene, Joshua. "Lattices, graphs, and Conway mutation." MR3049933.
  • [GS] Gompf, Robert; Stipsicz; András. Four-manifolds and Kirby Calculus.
  • [Ha] Hatcher, Allen, Notes on Basic 3-Manifold Topology. Available here.
  • [HKK] Hu, Po; Kriz, Daniel; Kriz, Igor. "Field theories, stable homotopy theory, and Khovanov homology." MR3465966.
  • [HKL] Hom, Jennifer; Karakurt, Çağrı; Lidman, Tye. "Surgery obstructions and Heegaard Floer homology." MR3548466.
  • [Hom] Hom, Jennifer. "Lecture notes on Heegaard Floer homology." Draft available in Canvas.
  • [Ja] Jacobsson, Magnus. "An invariant of link cobordisms from Khovanov homology." MR2113903.
  • [Kh1] Khovanov, Mikhail. "A categorification of the Jones polynomial." MR1740682.
  • [Kh2] Khovanov, Mikhail. "A functor-valued invariant of tangles." MR1928174.
  • [Kh3] Khovanov, Mikhail. "An invariant of tangle cobordisms." MR2171235.
  • [KM] Kronheimer, Peter; Mrowka, Tomasz. "Khovanov homology is an unknot-detector." MR2805599.
  • [KMOS] Kronheimer, Peter; Mrowka, Tomasz; Ozsváth, Peter; Szabó, Zoltán. "Monopoles and lens space surgeries." MR2299739.
  • [KS] Kirby, Robion; Scharlemann, Martin. "Eight faces of the Poincaré homology 3-sphere." MR0537730.
  • [L] Lipshitz, Robert. "A cylindrical reformulation of Heegaard Floer homology." MR2240908.
  • [Lee] Lee, Eun Soo. "An endomorphism of the Khovanov invariant." arXiv:math/0210213.
  • [Lev] Levine, Jerome. "Knot cobordism groups in codimension two." MR0246314.
  • [Lic] Lickorish, Raymond. An Introduction to Knot Theory.
  • [Liv1] Livingston, Charles. "A Survey of Classical Knot Concordance." MR2179265.
  • [Liv2] Livingston, Charles, Introduction to Knot Concordance (Work in Progress). Available here.
  • [LLS1] Lawson, Tyler; Lipshitz, Robert; Sarkar, Sucharit. "Khovanov homotopy type, Burnside category, and products." arXiv:1505.00213.
  • [LLS2] Lawson, Tyler; Lipshitz, Robert; Sarkar, Sucharit. "The cube and the Burnside category." MR3611723.
  • [LNS] Lipshitz, Robert; Ng, Lenhard; Sarkar, Sucharit. "On transverse invariants from Khovanov homology." MR3392962.
  • [LS1] Lipshitz, Robert; Sarkar, Sucharit. "A Khovanov stable homotopy type." MR3230817.
  • [LS2] Lipshitz, Robert; Sarkar, Sucharit. "A Steenrod square on Khovanov homology." MR3252965.
  • [LS3] Lipshitz, Robert; Sarkar, Sucharit. "A refinement of Rasmussen's s-invariant." MR3189434.
  • [LS4] Lipshitz, Robert; Sarkar, Sucharit. "Spatial refinements and Khovanov homology." MR3966803. Available here.
  • [LS5] Lipshitz, Robert; Sarkar, Sucharit. "Khovanov homology also detects split links." arXiv:1910.04246.
  • [McC] McCoy, Duncan. "Alternating knots with unknotting number one." MR3570147.
  • [MOSzT] Manolescu, Ciprian; Ozsváth, Peter; Szabó, Zoltán; Thurston, Dylan. "On combinatorial link Floer homology." MR2372850.
  • [MP] Miller, Allison N.; Piccirillo, Lisa. "Knot traces and concordance." MR3784230.
  • [Mu1] Murasugi, Kunio. "On periodic knots." MR0292060.
  • [Mu2] Murasugi, Kunio. "Jones polynomials of periodic links." MR0922222.
  • [Na] Naot, Gad. "The universal Khovanov link homology theory." MR2263052.
  • [NOT] Ng, Lenhard; Ozsváth, Peter; Thurston, Dylan. "Transverse knots distinguished by knot Floer homology." MR2471100.
  • [NT] Ng, Lenhard; Thurston, Dylan. "Grid diagrams, braids, and contact geometry." MR2500576.
  • [NW] Ni, Yi; Wu, Zhongtao. "Cosmetic surgeries on knots in S3." MR3393360.
  • [OSSz] Ozsv├íth, Peter; Stipsicz, András; Szabó, Zoltán. "Grid homology for knots and links." MR3381987. Available here.
  • [OSz1] Ozsváth, Peter; Szabó, Zoltán. "Holomorphic disks and topological invariants for closed three-manifolds." MR2113019.
  • [OSz2] Ozsváth, Peter; Szabó, Zoltán. "Holomorphic disks and knot invariants." MR2065507.
  • [OSz3] Ozsváth, Peter; Szabó, Zoltán. "Knot Floer homology and the four-ball genus." MR2026543.
  • [OSz4] Ozsváth, Peter; Szabó, Zoltán. "Holomorphic disks and three-manifold invariants: properties and applications." MR2113020.
  • [OSz5] Ozsváth, Peter; Szabó, Zoltán. "Holomorphic triangles and invariants for smooth four-manifolds." MR2222356.
  • [OSz6] Ozsváth, Peter; Szabó, Zoltán. "Holomorphic disks and genus bounds." MR2023281.
  • [OSz7] Ozsváth, Peter; Szabó, Zoltán. "Knot Floer homology and integer surgeries." MR2377279.
  • [OSz8] Ozsváth, Peter; Szabó, Zoltán. "Knot Floer homology and rational surgeries." MR2764036.
  • [OSz9] Ozsváth, Peter; Szabó, Zoltán. "On knot Floer homology and lens space surgeries." MR2168576.
  • [OSz10] Ozsváth, Peter; Szabó, Zoltán. "The Dehn surgery characterization of the trefoil and the figure eight knot." MR3956375.
  • [OSz11] Ozsváth, Peter; Szabó, Zoltán. "On the Heegaard Floer homology of branched double-covers." MR2141852.
  • [OSz12] Ozsváth, Peter; Szabó, Zoltán. "Heegaard diagrams and holomorphic disks." MR2102999. Available here.
  • [OSz13] Ozsváth, Peter; Szabó, Zoltán. "An introduction to Heegaard Floer homology." MR2249247. Available here.
  • [OSzT] Ozsváth, Peter; Szabó, Zoltán; Dylan Thurston. "Legendrian knots, transverse knots and combinatorial Floer homology." MR2403802.
  • [Pe] Perutz, Timothy. "Hamiltonian handleslides for Heegaard Floer homology." MR2509747.
  • [Pl] Plamenevskaya, Olga. "Transverse knots and Khovanov homology." MR2250492.
  • [Poli] Politarczyk, Wojciech. "Equivariant Khovanov homology of periodic links." MR4029632.
  • [Poly] Polyak, Michael. "Minimal generating sets of Reidemeister moves." MR2733246.
  • [Pu] Purcell, Jessica. Hyperbolic Knot Theory. Available here.
  • [Ra1] Rasmussen, Jacob. "Khovanov homology and the slice genus." MR2729272.
  • [Ra2] Rasmussen, Jacob. "Floer homology and knot complements." arXiv:math/0306378
  • [Sar] "Grid diagrams and the Ozsváth-Szabó tau-invariant." MR2915478.
  • [Se] Seed, Cotton. "Computations of the Lipshitz-Sarkar Steenrod Square on Khovanov Homology." arXiv:1210.1882.
  • [SW] Sarkar, Sucharit; Wang, Jiajun. "An algorithm for computing some Heegaard Floer homologies."MR2630063.
  • [Sza] Szabó, Zoltán. "A geometric spectral sequence in Khovanov homology." MR3431667.
  • [SZ] Stoffregen, Matthew; Zhang, Melissa. "Localization in Khovanov homology." arXiv:1810.04769.
  • [Te] Teichner, Peter. "Slice knots: knot theory in the 4th dimension." Available here.
  • [Zh] Zhang, Melissa. "A rank inequality for the annular Khovanov homology of 2-periodic links." arXiv:1707.03279.

Week-by-week list of references.

  1. Knot theory basics.
    • [Lic] Chapters 1, 2, 11: Knot diagrams etc. Seifert surfaces. Knot group and properties.
    • [Cr] Chapters 3, 4, 5: Similar material to [Lic], somewhat gentler exposition.
    • [BJ] An efficient introduction to smooth topology, including the isotopy extension theorem.
    • [Poly] Minimal sets of Reidemeister moves.
    • [Fr] Thurston geometries, geometrization theorem.
    • [Pu] Hyperbolic geometry and its applications to knot theory.
    • [Ha] Proofs of basic results in combinatorial 3-manifold topology: decomposition theorems, the sphere theorem, etc.
    • [Liv1], [Liv2], [Te] Knot concordance and slice knots.
    • [Lev] Levine paper about slice knots mentioned in class.
    • [CD] Proof that this strategy does not always work, and more references.
  2. Alexander and Jones polynomials. [L] chapters 3-8, 15, and references therein.
  3. Khovanov homology. [Kh1] is my preferred reference. Some people prefer [BN1].
  4. Khovanov homology for tangles and cobordisms.
    • [Kh2] Khovanov's tangle invariant.
    • [BN2] Bar-Natan's "picture-world" invariant of tangles and cobordisms.
    • [Kh3] Khovanov's proof of functoriality under cobordisms, using his tangle invariant.
    • [Ja] The original proof that Khovanov homology is functorial up to sign.
    • [Na] An analysis of exactly what information Bar-Natan's invariant contains.
    • [Bl, Ca, CMW]. Further discussion of the sign in the cobordism maps on Khovanov homology.
  5. Lee spectral sequence, Rasmussen's s-invariant, and the slice genus.
    • [Lee] The original paper introducting the Lee spectral sequence.
    • [Ra1] Rasmussen's paper defining the s-invariant and proving its basic properties.
    • [GS] See pp. 377 and 522 for a proof that existence of topologically but non-smoothly slice knots implies the existence of exotic smooth structures on R4.
    • [MP] More discussion of the 0-trace and its relation to concordance.
  6. The Khovanov stable homotopy type.
    • [LS1] Original construction of the Khovanov stable homotopy type. This is not the construction we are discussing in class.
    • [HKK] Alternate construction. [LLS1] Proof that these constructions agree. Probably the closest reference to our treatment in class.
    • [LS2], [Se] Computation of the induced map sq2 on Khovanov homology. Families of knots with isomorphic Khovanov homology which are distinguised by sq2.
    • [LS3] Refinement of the Rasmussen invariant using sq2.
    • [LLS2], [LS4] Survey papers on this material.
    • Periodic knots: [Mu1], [Mu2] Murasugi's conditions on the Alexander and Jones polynomials of periodic knots. [Ch], [Poli], [BP], [Cor], [Zha], [SZ], [BPS]: results about periodic knots and Khovanov homology.
  7. Grid diagrams and knot Floer homology.
    • [MOSzT], [OSSz]: where I am drawing definitions and conventions from, mostly. [MOSzT] is the first combinatorial proof of invariance of knot Floer homology.
    • [MOS]: the origin of this construction of knot Floer homology. [OSz2], [Ra2]: the original definition of knot Floer homology (discussed in week 8).
    • [Sar]: a combinatorial proof of the basic properties of the tau invariant (also found in [OSSz]). [OSz3]: original definition of tau.
    • [Et]: Introduction to Legendrian and transverse knots.
    • [OSzT]: construction of Legendrian and invariants from grid diagrams. [NT]: more on the relationship between grid diagrams and contact topology. [NOT]: more knots distinguished by these invariants.
    • [Pl]: transverse knot invariants from Khovanov homology. [LNS]: more about these invariants and some (perhaps uninteresting) refinements of them.
  8. Heegaard diagrams and holomorphic disks.
    • [OSz1]: the original definition of Heegaard Floer homology.
    • [L]: a reformulation of this construction.
    • [Pe]: Explicit construction of symplectic forms on the symmetric products of a Riemann surface, so that the Heegaard tori are Lagrangian.
    • [SW]: a construction of nice diagrams, a kind of analogue of grid diagrams, for general 3-manifolds.
    • Some survey articles: [Hom], [OSSz], [OSz12], [OSz13].
  9. Surgery formulas and applications.
    • [OSz2], [Ra2]: Original definition of knot Floer homology, large surgery formula.
    • [OSz4] Surgery exact triangle.
    • [OSz5] Absolute grading.
    • [OSz6] Genus detection.
    • [OSz7], [OSz8]: Integer and rational surgery formulas.
    • Applications mentioned: [KMOS], [OSz9], [OSz10], [ZW], [Gr1], [Gr2], [McC], [HKL].
    • [KS] Equivalent descriptions of the Poincaré sphere.
  10. The Ozsváth-Szabó and Kronheimer-Mrowka spectral sequences and some detection results.
    • [OSz11] Ozsváth-Szabó spectral sequence.
    • [Sza] Szabó spectral sequence, a conjectured combinatorial analogue of [OSz11].
    • [KM] Khovanov homology detects the unknot.
    • [LS5] Khovanov homology detects split links. Hopefully comprehensive references to other Khovanov homology detection results in the introduction.