Knot Linking

In general, it is possible to link two -D Hyperspheres in -D space in an infinite number ofinequivalent ways. In dimensions greater than in the piecewise linear category, it is true that these spheres arethemselves unknotted. However, they may still form nontrivial links. In this way, they are something like higher dimensionalanalogs of two 1-spheres in 3-D. The following table gives the number of nontrivial ways that two -DHyperspheres can be linked in -D.

Two 10-D Hyperspheres link up in 12, 13, 14, 15, and 16-D, then unlink in 17-D, link up again in18, 19, 20, and 21-D. The proof of these results consists of an ``easy part'' (Zeeman 1962) and a ``hard part'' (Ravenel1986). The hard part is related to the calculation of the (stable and unstable) Homotopy Groups of Spheres.
References

Bing, R. H. The Geometric Topology of 3-Manifolds. Providence, RI: Amer. Math. Soc., 1983.

Ravenel, D. Complex Cobordism and Stable Homotopy Groups of Spheres. New York: Academic Press, 1986.

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 7, 1976.

Zeeman. ``Isotopies and Knots in Manifolds.'' In Topology of 3-Manifolds and Related Topics (Ed. M. K. Fort). Englewood Cliffs, NJ: Prentice-Hall, 1962.

• Directed Graph
• Directional Derivative
• Direction Cosine
• Directly Similar
• Director Curve
• Direct Product (Group)
• Direct Product (Matrix)
• Direct Product (Set)
• Direct Product (Tensor)
• Directrix (Conic Section)
• Directrix (Graph)
• Directrix (Ruled Surface)
• Direct Search Factorization
• Direct Sum (Module)
• Dirichlet Beta Function
• Dirichlet Boundary Conditions
• Dirichlet Conditions
• Dirichlet Divisor Problem
• Dirichlet Energy
• Dirichlet Eta Function
• Dirichlet Fourier Series Conditions
• Dirichlet Function
• Dirichlet Integrals
• Dirichlet Kernel
• Dirichlet Lambda Function