Oberseminar Algebra und Geometrie
Summer term 2023: Knot theory and quandles
Chaired by I. Halupczok, H. Kammeyer and B. Klopsch.
Organised by H. Kammeyer
All talks take place on Fridays at 12:30 pm in 25.22.03.73.
If you want to participate but did not yet receive any mails related to the seminar, please get in touch with I. Halupczok so that your address is added to the mailing list.Aims and Content - a short description
This term's topic will be "Knot theory and quandles". We will begin with an introduction to knot theory covering the foundations (knots and knot equivalences, Reidemeister moves, ...) and standard concepts (e.g. the knot group, Seifert surfaces and the Alexander polynomial). We will then turn our attention to the less well-known but all the more powerful notion of knot quandles. These can be described as an algebraic concept that better satisfies the needs of knot theory than the knot group does. In particular, we will show that the knot quandle is a complete invariant: it distinguishes all inequivalent knots. For a more detailed overview of the topic, please refer to the seminar program.Schedule
(tba)07.04.23 | 1. No Seminar - (Good Friday) |
14.04.23 | 2. Knots and knot equivalence (Margherita Piccolo)
Define knots (and links), discuss wild and tame knots, state that we restrict to the latter ones, introduce (ambient) isotopies, knot equivalences (leave out delta moves if you need to save time), knot projections, Reidemeister moves and discuss the characterisation of knot equivalences in terms of Reidemeister moves. (Burde-Zieschang, Chapter 1.A-C). |
21.04.23 | 3. Geometric concepts (Giada Serafini)
Define Mirror images, invertible and amphicheiral knots, alternating knot projections, Seifert surfaces and genus, meridian and longitude, knot sums (sometimes called products), companion and satellite knots and discuss tricolorability as an elementary invariant of knots. (Burde-Zieschang, Chapter 2.A-C and Adams, Chapter 1.5 for tricolorability). |
28.04.23 | 4. Knot group and homology (Daniel Echtler)
Discuss the homology of the knot complement, the Wirtinger Presentation of the knot group, define companion and satellite knots and discuss the knot groups of satellite and companion knots. (Burde-Zieschang, Def. 2.8 and Chapter 3.A and 3.B). |
05.05.23 | 5. Peripheral systems (Luca Leon Happel)
Discuss peripheral systems, Waldhausen's Theorem and the characterisation of invertibility and amphicheirality. Finally, show the asphericity of the knot complement. (Burde-Zieschang, Chapter 3.C and 3.F). |
12.05.23 | 6. The Alexander Polynomial (Jan Hennig)
Discuss cyclic knot coverings and the Alexander module, band projections, Seifert matrices, Alexander matrices, Alexander polynomials and their properties. (Burde-Zieschang, Chapter 8.A-D. |
19.05.23 | 7. Fox differential calculus (Daniel Dratschuk)
Discuss the homology of covering spaces, Fox differential calculus and the calculation of Alexander matrices and polynomials. (Burde-Zieschang, Chapter 9.A-C) |
26.05.23 | 8. Introduction to quandles (Rebecca Lenger)
Introduce quandles, abelian quandles, conjugation quandles, Alexander quandles, involutary quandles (Kei), the core, homomorphisms of quandles, free quandles, show that free quandles arise as subquandles of free groups, discuss admissible equivalence relations, presentations of quandles and the adjunction. Stress the non-injectivity of the unit of the adjunction. (Joyce, Sections 1 and 2.1 and Nardin for Alexander quandles, free quandles, admissible equivalence relations and presentations.) |
02.06.23 | 9. Coset quandles and augmented quandles (Martina Conte)
Define coset quandles and homogeneous quandles and show that the latter are coset quandles. State the generalization to general quandles and discuss augmented quandles, their homomorphisms, limits and colimits and finally, their quotients arising from normal subgroups. (Joyce, Sections 2.4, 2.10 and 2.11). |
09.06.23 | 10. Fundamental quandle of a pair of spaces (Moritz Petschick)
Introduce the fundamental quandle of a pair of spaces. Instead of copying the uninformative formulas, draw informative pictures (of nooses, the action of loops on nooses,...). Follow the formal viewpoint as you find suitable. Show that the fundamental quandle of a disk with respect to the center point gives essentially the winding number. (Joyce, Sections 4.1 and 4.2) |
16.06.23 | 11. Seifert-van Kampen for quandles (Holger Kammeyer)
Show the Seifert--van Kampen theorem for quandles. To do so, it might be helpful to recall the categorical concept of a pushout and van Kampen's theorem for fundamental groups of spaces with open covers. As examples, consider punctured surfaces. (Joyce, Sections 4.3 and 4.4). |
23.06.23 | 12. Knot quandles and diagram quandles (Thor Wittich)
Give the geometric defintion of a knot quandle as the subquandle of the fundamental quandle of nooses with winding number one. Explain how to obtain a knot quandle from a Wirtinger presentation of the knot group. Explain briefly that the latter is a knot invariant using Reidemeister moves. Using van Kampen's theorem, show that the two definitions give isomorphic quandles. (Joyce, Sections 4.5-4.8). |
30.06.23 | 13. No Oberseminar |
07.07.23 | 14. Completeness and abelian knot quandles as Alexander invariants (Immanuel Halupczok)
Show that the knot quandle is a complete knot invariant by showing that it encodes the peripheral subgroup structure. Show that the Alexander invariant and the abelian knot quandle determine each other. If time permits, mention cyclic knot invariants and involutory knot quandles. (Joyce, Sections 4.9-4.12). |
14.07.23 | Discussion of topics for the next semester (all of us) |
Literature
For the references above, please refer to the seminar program.Archive
WS 2022/23: Combinatorics and Commutative Algebra
SS 2022: Discrete Groups, Expanding Graphs and Invariant Measures
WS 2021/22: Superrigidity
SS 2021: Group cohomology
SS 2020 and WS 20/21: cancelled due to pandemic
WS 2019/20: Intersection theory
SS 2019: Knots and primes
WS 2018/19: The Grothendieck group of varieties and stacks
SS 2018: Arithmetic Groups - Basics and Selected Applications
WS 2017/18: Algebraic K-theory
SS 2017: Berkovich spaces
WS 16/17: Resolution of singularities and alterations
SS 2016: Modular Representation Theory
WS 15/16: The Milnor Conjectures
SS 2015: Rationality
WS 14/15: Essential Dimension
SS 2014: Varieties of Representations
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