$\def\bbN{{\mathbb N}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calT{{\mathcal T}}$
 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;

© | Dror Bar-Natan: Classes: 2022-23:

## Fast Computations in Knot Theory

### Tsuda University, June-July 2023

https://drorbn.net/23-FC

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).

Day 1 - Thursday June 29, 9:30-12 and 1-2:30.
HW1 was assigned and was due on Monday July 3.
Day 2 - Friday June 30, 9:30-12.
• 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.

Day 3 - Monday July 3, 9:30-12 and 1:30-2:30.
• Many many preliminaries: direct sums, tensor products, complexes, homology, functoriality, Euler characterics, and making everything graded.
• Homology of spaces.
Today's handout: KH4Knots.pdf
Day 4 - Wednesday July 5, 9:30-12.
• Khovanov homology: The definition.
HW2 was assigned and was due on Monday July 10.
Solution of HW2 Problem 2: HW2Problem2.pdf, HW2Problem2.nb.
Day 5 - Friday July 7, 9:30-12 and 1-2:30.
• How to prove things about complexes?
• Khovanov homology: Invariance.
• Khovanov homology: Computation.
Today's handout: FirstKHProgram.pdf
Today's Mathematica notebook: FirstKHProgram@.pdf, FirstKHProgram@.nb.
Day 6 - Monday July 10, 9:30-12.
• 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.
Knots in Three and Four Dimensions as in Cornell University in 2015.
HW3 is now online and is due on Thursday July 13.
References.
1. Dror Bar-Natan, On Khovanov's Categorification of the Jones Polynomial, Algebraic and Geometric Topology 2-16 (2002) 337-370.
2. Dror Bar-Natan, Khovanov's Homology for Tangles and Cobordisms, Geometry and Topology 9-33 (2005) 1443-1499.
3. Dror Bar-Natan, Fast Khovanov Homology Computations, Journal of Knot Theory and Its Ramifications, 16-3 (2007) 243-255.
4. Allen Hatcher, Algebraic Topology.
5. Mikhail Khovanov, A Categorification of the Jones Polynomial, Duke Math. J. 101 (2000), no. 3, 359-426.
6. Erica Klarreich, Graduate Student Solves Decades-Old Conway Knot Problem, Quanta Magazine on May 19 2020.
7. Louis H. Kauffman, "On Knots", Princeton University Press 1988.
8. W. B. Raymond Lickorish, "An Introduction to Knot Theory", GTM 175, Springer 1997.
9. Tomotada Ohtsuki, 結び目の不変量 (Invariants of Knots), Kyoritsu Shuppan, 2015.
10. Lisa Piccirillo, The Conway knot is not slice, Ann. of Math. (2) 191(2): 581-591 (March 2020).02923.

Further resources.