Mathematics Colloquia and Seminars

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Just as closed curves can be knotted and linked in 3-dimensional space, 2-spheres can be knotted and linked in 4-space. However, the latter phenomenon is not completely analogous to the former. In particular, if two disjoint spheres embedded in R^4 each have only a single maximum and minimum point (and no saddle points), then they must be unlinked. This is in stark contrast with the case of classical links, as illustrated by the Hopf link. Nonetheless, there are non-trivial two-component sphere links in R^4 in which each component is unknotted. In this talk, I'll discuss what it means for spheres to be knotted and linked in 4-space, how such spheres can be constructed, and what lies behind these results.