Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Motivic Knot Theory: Quadratic Linking Degree

Description

This lecture by Clémentine Lemarié-Rieusset introduces the quadratic linking degree, a concept in motivic knot theory that parallels the linking number in algebraic geometry. The lecture covers knot theory basics, oriented links in algebraic geometry, Chow groups, intersection theory, and Milnor-Witt K-theory. The quadratic linking degree is defined as the image of the intersection product of Seifert surfaces by the boundary map. Examples like the Hopf link and the Solomon link are discussed to illustrate the theory. The lecture concludes with a presentation of the preprint 'The Quadratic Linking Degree' by Clémentine Lemarié-Rieusset.

Official source

Related concepts (31)

Quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K. If , and the quadratic form equals zero only when all variables are simultaneously zero, then it is a definite quadratic form; otherwise it is an isotropic quadratic form.

Quadratic equation

In algebra, a quadratic equation () is any equation that can be rearranged in standard form as where x represents an unknown value, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.) The numbers a, b, and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term.

Intersection theory

In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form. There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov-Witten theory and the extension of intersection theory from schemes to stacks.

Chow group

In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product.

Intersection number

In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem. The intersection number is obvious in certain cases, such as the intersection of the x- and y-axes in a plane, which should be one.

Related lectures (32)

Homology of Riemann Surfaces

Explores the homology of Riemann surfaces, including singular homology and the standard n-simplex.

Differential Forms on Manifolds

Introduces differential forms on manifolds, covering tangent bundles and intersection pairings.

Varieties with nef anti-canonical: Surjective Albanese

Presents a proof that smooth projective varieties with nef anti-canonical divisor have surjective Albanese morphism.

Conformal Blocks: Functions on Moduli Spaces

Covers conformal blocks and their significance in complex structures and moduli spaces.

Knot Theory: The Quadratic Linking Degree Log in to Mediaspace to watch this video