2026/2027
Контактная топология узлов: от дифференциальных градуированных алгебр Чеканова–Элиашберга к полному инварианту узлов
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Язык:
английский
Кредиты:
6
Контактные часы:
60
Course Syllabus
Abstract
"A knot is a closed curve in $\mathbb{R}^3$ without self-intersections; two knots are equivalent if one can be smoothly deformed into the other. The central problem of knot theory is distinguishing knots: how can one prove that a given knot cannot be deformed into another? For this one uses invariants -- quantities or algebraic structures assigned to a knot that do not change under deformation. Many invariants are known (the Alexander polynomial, the Jones polynomial, the knot group on its own), but most are incomplete: they fail to distinguish certain non-equivalent knots. A classical complete invariant -- the knot group together with the peripheral subgroup -- is a purely topological construction. In 2018, Ekholm, Ng, and Shende discovered a complete invariant of a very different nature, coming from contact geometry.
Contact geometry has its origins in geometric optics and classical mechanics and studies certain natural geometric structures on odd-dimensional manifolds. Over the last decades its methods have found unexpected applications to topology. In particular, to a knot $K \subset \mathbb{R}^3$ one canonically associates a two-dimensional surface (a torus) inside a certain five-dimensional contact manifold; choosing a point off the knot determines an additional sphere in the same manifold. Counting holomorphic curves with boundaries on these submanifolds produces an algebraic structure -- the KCH-triple equipped with a product map -- from which the knot can be completely recovered.
No prior knowledge of knot theory or contact geometry is assumed; all necessary background is developed from scratch. The course covers: the knot group and the peripheral subgroup; fundamentals of symplectic and contact geometry; the Chekanov–Eliashberg differential graded algebra and its augmentations; augmentation varieties and the $A$-polynomial; knot contact homology; the cord algebra; and enhanced knot contact homology. The proof of the completeness theorem is the final goal of the course."
Learning Objectives
- Formation of a systematic understanding of classical knot theory, contact geometry, and Legendrian knot invariants used in the course.
- Development of a unified view of the Chekanov-Eliashberg DGA, augmentations, knot contact homology, and their relations to classical knot invariants.
- Formation of an understanding of how enhanced knot contact homology recovers the knot group and peripheral structure, leading to the Ekholm-Ng-Shende completeness theorem.
Expected Learning Outcomes
- Students can compute knot groups from knot diagrams using Wirtinger presentations and can identify the meridian, longitude, and peripheral subgroup in basic examples.
- Computes the Thurston-Bennequin and rotation numbers from front and Lagrangian projections and distinguishes standard Legendrian examples using these invariants.
- Constructs the Chekanov-Eliashberg DGA for a given simple Lagrangian projection, computes its differential, and verifies the DGA relations in explicit examples.
- Finds augmentations of explicit DGAs and computes the corresponding linearized differential and linearized contact homology.
- Constructs the algebraic equations defining basic character and augmentation varieties and applies elementary elimination methods to recover A-polynomial factors in standard examples.
- Describes the knot contact homology DGA of a conormal torus and identifies its degree zero part with the cord-algebra description used to relate contact homology to the knot group.
- Defines the two-component Legendrian link associated with a knot and an exterior point, constructs the filtration by mixed Reeb chords, and identifies the three components and module structures of the KCH-triple.
- Computes and interprets the product structures used in enhanced knot contact homology and matches the KCH product with the corresponding Pontryagin product in the broken-string model.
- Reconstructs the knot group and its peripheral data from the KCH-triple and reproduces the main algebraic and topological steps of the Ekholm-Ng-Shende completeness proof, using the geometric comparison theorems stated in the course.
Course Contents
- Classical Knot Theory and Foundations of Contact Geometry
- Legendrian Knots and the Chekanov-Eliashberg DGA
- Augmentations, Linearized Contact Homology, and the A-Polynomial
- Knot Contact Homology and the KCH-Triple
- String Topology and the Completeness Theorem
Assessment Elements
- Homework 1Written homework on classical knot theory, Legendrian knots, and the Chekanov-Eliashberg DGA.
- Homework 2Written homework on augmentations, linearized contact homology, and augmentation varieties.
- Homework 3Written homework on knot contact homology, the cord algebra, and the KCH-triple.
- Final ExamWritten final examination covering the main topics of the course.
Interim Assessment
- 2026/2027 2nd module0.2 * Homework 2 + 0.2 * Homework 1 + 0.2 * Homework 3 + 0.4 * Final Exam
Bibliography
Recommended Core Bibliography
- Allen Hatcher. (2002). Algebraic topology. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.87FE219C
- David A. Cox, John Little, Donal O’Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. - Springer International Publishing, Switzerland, 2015. Print ISBN: 978-3-319-16720-6. Online ISBN: 978-3-319-16721-3.
- Dusa McDuff, & Dietmar Salamon. (2017). Introduction to Symplectic Topology: Vol. 3rd ed. OUP Oxford.
- Loring W. Tu. (2010). An Introduction to Manifolds (Vol. 2nd ed. 2011). Springer.
Recommended Additional Bibliography
- Chmutov, S., Duzhin, S., & Mostovoy, J. (2011). Introduction to Vassiliev Knot Invariants. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1103.5628