Seminar Algebra and Geometry: Johan Björklund (Institut de Mathématiques de Jussieu)

Lifting real algebraic curves to real algebraic knots

A real algebraic rational knot is an algebraic embedding of CP¹ into CP³ by real polynomials. By applying a real projection to a (generic) projective plane we obtain a real rational planar curve of the same degree, which should be thought of as the associated knot diagram (with appropriate decorations at double points). In classical smooth knot theory, knots and their diagrams are closely interconnected. For each decorated diagram, there is a corresponding knot (constructed using an appropriate height-function). In the algebraic setting this does not hold true, there exists decorated real algebraic planar curves which can not be lifted to an appropriate knot.

In this talk I will explain how to associate a hyperplane arrangement to a nodal planar rational real algebraic curve. This arrangement describes the space of nonsingular liftings and allows us to calculate the homology (and thus, in particular, the number of liftings up to rigid isotopy). We will show that, up to degree 5, this hyperplane arrangement is a rigid isotopy invariant (of planar curves) and can provide real algebraic analogues of the classical Reidemeister moves. Obstructions in the case of higher degrees will be discussed. The talk should be accessible to nonspecialists.

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