Topics in Topology 2021
This is the homepage for the seminar course Topics in Topology, aimed at MSc students. We plan to study grid homology and knot Floer homology if time allows. Students from Dutch universities other than Groningen are also entitled to take this course and have a recognition of credits (5 EC) at their home universities.
We meet on Mondays 15:00-17:00, of course online these days. If it clashes with another course please let us know.
For more information about the seminar please see the course description below or contact us.
Organisers: Roland van der Veen and Jorge Becerra
Schedule
We kindly ask students to select a talk and let us know your choice. OP=OP!
The references below refer to the book Grid Homology for Knots and Links by Ozsváth, Stipsicz and Szabó.
Monday 1 February 2021, 15:00 - 17:00 via this link
15:00 - 15:20: (Introduction Roland/Jorge) Knot isotopy, examples and Seifert surfaces.
15:30 - 16:20: Aarnout Los: The Alexander polynomial (sec. 2.3 and 2.4, especially 2.4.1 and 2.4.6). Aarnout's notes.
16:30 - 17:00: Exercises & discussion.
Monday 8 February 2021, 15:00 - 17:00 via this link
15:00 - 15:20: (Introduction R/J) Grid diagrams, stabilisation and commutation. Slides.
15:30 - 16:20: Oscar Koster: The Alexander polynomial from grid diagrams (sec. 3.3, especially 3.3.6). Oscar's slides and notes.
16:30 - 17:00: Exercises & discussion. Exercise Hopf link solved.
Monday 15 February 2021, 15:00 - 17:00 via this link
15:00 - 15:20: (Introduction R/J) Grid states, bigrading and rectangle. Notes.
16:30 - 17:00: Exercises & discussion.
Monday 22 February 2021, 15:00 - 17:00 via this link
15:00 - 15:20: (Introduction R/J) Homological algebra of doubly graded chain complexes and more on grid homology.
15:30 - 16:20: Robbert Scholtens: Examples (sec. 4.8, especially 4.8.1 and 4.8.2). Robbert's notes, presentation and the .py file to compute the bigrading.
16:30 - 17:00: Exercises & discussion.
Monday 1 March 2021, 15:00 - 17:00 via this link
15:00 - 15:20: (Introduction R/J) Gap filling on homological algebra. Notes
15:30 - 16:20: Sven Bootsma: Invariance of grid homology: stabilisation (subsec. 5.2.1, giving a self-contained proof of 5.2.2-5.2.3). Sven's notes and slides.
16:30 - 17:00: Exercises & discussion.
Monday 8 March 2021, 15:00 - 17:00 via this link
15:00 - 15:20: (Introduction R/J) More on the Alexander polynomial.
15:30 - 16:20: Roelien Smit: The Alexander polynomial from grid homology (sec. 4.7, especially 4.7.6). Roelien's notes and slides.
16:30 - 17:00: Exercises & discussion.
Monday 15 March 2021, 15:00 - 17:00 via this link
15:00 - 15:20: (Introduction R/J) Some properties of grid homology. Notes
15:30 - 16:20: Kevin van Helder: A sharp knot genus bound (sec. 3.4 and 7.2, especially 3.4.11 and 7.2.2). Kevin's notes and slides.
16:30 - 17:00: Exercises & discussion.
Monday 22 March 2021, 15:00 - 17:00 via this link
15:00 - 15:20: (Introduction R/J) The holomorphic theory.
15:30 - 16:20: Jeffrey Weenink: Knot Floer homology and its relation with grid homology. Jeffrey's notes and slides.
16:30 - 17:00: Exercises & discussion.
Course description
In this seminar course on grid homology, participants are expected to give at least one 45-minute talk, possibly more depending on the number of students, about a topic of their choice among the proposed ones. They must also elaborate a detailed handout about their talk, to be brought along to the meeting. The handout should contain (most of) the proofs of the statements. Students are also required to prepare some exercises to have a joint discussion after their talk. Attendance to every meeting is required. We urge participants to stick to 45 minutes during their talks (for that some good advice is to rehearse the talk and time oneself).
Prerequisites
Students should be familiar with basic concepts of topological spaces and commutative algebra. Knowledge from the courses Introduction to metric and topological spaces, linear algebra, group theory and multivariable analysis will suffice.
Evaluation
The final grade will be based on your presentation (50%), your handout (25%) and your participation in the joint discussions (25%).
References
Our main reference will be
Ozsváth, P. & Stipsicz, A. & Szabó, Z - Grid Homology for Knots and Links
Some other complementary references are
Davis, F.D. & Kirk, P. - Lecture notes in algebraic topology.
Hatcher, A. - Algebraic Topology (see p. 102-107 for a quick definition of homology).
Hom, J. - Heegaard-Floer homology
Manolescu, C. - An introduction to knot Floer homology.
Ozsváth, P. & Szabó, Z. - Heegaard Floer Homology and Knot Theory (Chapters 1 and 2)