Mathematics Colloquia and Seminars

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Just as closed curves in 3-dimensional space can be knotted in different ways, there are may inequivalent ways to embed a closed surface into 4-dimensional space. The study of such surface embeddings and their relation to standard knot theory goes back to the 1920's, when Artin described the construction of certain knotted 2-spheres in R^4 and examined the classical knots that arise as "slices" of such spheres. More generally, every knot can be realized as such a "slice" of some embedded surface in R^4. In this way, knot theorists are led to study isotopy classes of embedded surfaces in R^4. Such embeddings can be difficult to visualize, but one can conduct such a study by using methods analogous of those classical knot theory. In particular, any isotopy of a closed surface S in R^4 can be realized as a sequence of Reidemeister-type moves defined for a suitable projection of S into a hyperplane. These moves (known as the Roseman moves) lead to several related combinatorial descriptions of surface isotopies. After describing the Roseman moves and sketching the Morse-theoretic proof of their completeness, I will overview two of these combinatorial methods and sketch some concrete calculations.