University of Pennsylvania, Department of Mathematics, Philadelphia, PA 19104
Carl Friedrich Gauss defined an integral formula for the linking number of two curves in R^3. Intuitively, the linking number measures the number of times the curves loop around one another, or to what extent they are “linked.” With this informal definition in mind, one would expect that the linking number is an integer topological invariant, and this assertion is true. Linking integrals are important in electrodynamics, plasma physics, and molecular biology. In the 1960’s, Calugareanu and White showed that if the two curves bound a ribbon in R^3, the linking number can be expressed as the sum of two separate contributions called the “writhe” and “twist.” Writhe is defined as the linking integral of one edge of the ribbon with itself. Twist, on the other hand, measures the extent to which the unit vector pointing across the ribbon twists around the edge. Although the writhe and twist of a ribbon are geometric invariants, they are not topological nor are they integer-valued in general. The object of our current research concerns extending the notions of link, writhe, and twist to higher dimensional Euclidean spaces. For example, the linking integral formula can be generalized to compute the linking number of two smooth, closed, compact surfaces in R^5. We prove the Calugareanu-White Theorem in this case and generalize to m-dimensional submanifolds in (2m + 1)-dimensional Euclidean space. It is perhaps surprising that when m is even, the writhing number is always zero, so that the twist integral is also an integral topological invariant. We show that the twist equals half of the Euler class of the m-dimensional subbundle of the normal bundle of the submanifold. We further generalize these results to the case where the ambient space is the (2m + 1)-dimensional sphere or hyperbolic space.
[Abstract (DOC)]