1. Atsuko Yamaguchi; 2. Haruko Miyazawa; 3. ?; 4. Aoi Wakuda; 5. ?; 6. Ayumu Inoue; 7. Toyo Taniguchi; 8. ?; 9. Naoki Kimura; 10. Dror Bar-Natan; 11. Takashi Hara; 12. ?; 13. ?; 14. Yusuke Kuno; 15. ?; 16. ?; 17. Gabi; |

**Tagline.** A half is better than a whole!

**Idea.** Do the computational side of Piccirillo's
"The Conway Knot is Not Slice", Ann. of
Math. (2) 191(2): 581-591 (March 2020), arXiv:1808.02923
(see also an article
in Quanta
Magazine).

**Course Purpose and Content / Learning Objectives.** Learn about
the Jones polynomial and about Khovanov homology, and how to compute
them, and how to use "tangles" to compute them even faster. Along the
way learn a bit about homology theory and about category theory. Actually
implement some of the algorithms learned!

**Preliminaries.** Absolute confidence with linear algebra:
vector spaces, linear transformations, kernels, images, Gaussian
elimination. Better if you know "tensor product" and "homology" even if
just barely.

**Reading Preliminaries.** Before the
start of the course you must read the Quanta
Magazine article (even without fully understanding
it), and you should skim through the Piccirillo paper.

**Evaluation Method.** Attendance (40%) and Homework (3 assignments, 20% each).

- A quick introduction to knot theory.

See Day1Gallery.pdf and Day1Gallery.html. - The Jones polynomial.
- Computing the Jones polynomial.

Today's Mathematica notebook: KauffmanBracket@.pdf, KauffmanBracket@.nb.

- A half is better than a whole: Computing the Jones polynomial much faster.

Today's Mathematica notebook: FasterJones@.pdf, FasterJones@.nb. - Cows are better than numbers! Complexes are not so bad either.

- Many many preliminaries: direct sums, tensor products, complexes, homology, functoriality, Euler characterics, and making everything graded.
- Homology of spaces.

- Khovanov homology: The definition.

Solution of HW2 Problem 2: HW2Problem2.pdf, HW2Problem2.nb.

- How to prove things about complexes?
- Khovanov homology: Invariance.
- Khovanov homology: Computation.

Today's Mathematica notebook: FirstKHProgram@.pdf, FirstKHProgram@.nb.

- Categories and complexes in a category.
- Homotopy in topology and in algebra.
- Khovanov homology for tangles.
- Formal Gaussian elimination and delooping.
- FastKh / a meta-half is better than a meta-whole.

HW3 is now online and is due on Thursday July 13.

- Dror Bar-Natan, On Khovanov's Categorification of the Jones Polynomial, Algebraic and Geometric Topology 2-16 (2002) 337-370.
- Dror Bar-Natan, Khovanov's Homology for Tangles and Cobordisms, Geometry and Topology 9-33 (2005) 1443-1499.
- Dror Bar-Natan, Fast Khovanov Homology Computations, Journal of Knot Theory and Its Ramifications, 16-3 (2007) 243-255.
- Allen Hatcher, Algebraic Topology.
- Mikhail Khovanov, A Categorification of the Jones Polynomial, Duke Math. J. 101 (2000), no. 3, 359-426.
- Erica Klarreich, Graduate Student Solves Decades-Old Conway Knot Problem, Quanta Magazine on May 19 2020.
- Louis H. Kauffman, "On Knots", Princeton University Press 1988.
- W. B. Raymond Lickorish, "An Introduction to Knot Theory", GTM 175, Springer 1997.
- Tomotada Ohtsuki, 結び目の不変量 (Invariants of Knots), Kyoritsu Shuppan, 2015.
- Lisa Piccirillo, The Conway knot is not slice, Ann. of Math. (2) 191(2): 581-591 (March 2020).02923.

- Previous knot theory classes that I've given: 273a - Knot Theory as an Excuse (Harvard, 1994), 273b - Knot Theory as an Excuse (Harvard, 1995), Seminar on Knots and Lie Algebras (Jerusalem, 1997), Three Dimensional Manifolds (Jerusalem, 1998), Knot Theory (Jerusalem, 2001), Knots and Feynman Diagrams (Jerusalem, 2001), Seminar on Knot Theory (Jerusalem, 2002), 1350F - Knot Theory (Toronto, 2003), 1350F - Algebraic Knot Theory (Toronto, 2006), 1352S - Algebraic Knot Theory, (Toronto, 2007), 1350F - Algebraic Knot Theory, (Toronto, 2009), The wClips Seminar, (Toronto, 2012), (u, v, and w knots) x (topology, combinatorics, low algebra, and high algebra), (Aarhus, 2013), 1350S - Algebraic Knot Theory, (Toronto, 2014), 1350S - Algebraic Knot Theory - Poly-Time Computations, (Toronto, 2017), 1350F - Topics in Knot Theory, (Toronto, 2020). 1350F - Topics in Knot Theory, (Toronto, 2021).
- My 23-FastComputations Pensieve Folder.