Knots and Surfaces in Real Algebraic and Contact Geometry
Sammanfattning: This thesis consists of a summary and three articles. The thesis is devoted to the study of knots and surfaces with additional geometric structures compared to the classical smooth structure.In Paper I, real algebraic rational knots in real projective space are studied up to rigid isotopy and we show that two real rational algebraic knots of degree at most 5 are rigidly isotopic if, and only if, their degree and encomplexed writhe are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree at most 6. Furthermore, an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented.In Paper II, we construct an invariant of parametrized generic real algebraic surfaces in real projective space which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: an intersection of two real sheets, an intersection of two complex conjugate sheets or a Whitney umbrella. The Brown invariant was expressed through a self linking number of the self intersection by Kirby and Melvin. We extend their definition of this self linking number to the case of parametrized generic real algebraic surfaces.In Paper III, we give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in the product of a punctured Riemann surface with the real line. As an application we show that for any nonzero homology class h, and for any integer k there exist k Legendrian knots all representing h which are pairwise smoothly isotopic through a formal Legendrian isotopy but which lie in mutually distinct Legendrian isotopy classes.
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