Mathematics Colloquia and Seminars

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We will consider low bounds for the number of lines meeting given 4 disjoint smooth closed curves in the real projective 3-space in a given cyclic order. Similarly, we will estimate the number of circles meeting in a given cyclic order 6 disjoint smooth closed curves in Euclidean 3-space. The estimations are formulated in terms of linking numbers of the curves and based on a study of the surface swept by projective lines meeting 3 given smooth closed curves and the surface swept by circles meeting 5 given disjoint smooth closed curves. The results admit generalisations to higher dimensions and more complicated patterns of intersections. Lines and circles can be replaced by configurations of curves of other kinds; linking numbers, by Vassiliev invariants.