Topology in Molecular Biology

10 Hopf Fibration and Its Applications 187

The article [13] has shown that there are infinitely many simple closed curves
onS^2 that are critical points forF. Therefore there are many embedded Hopf
tori inR^3 that are not a minimal torius inS^3.

Problem 2.Is it possible to construct “critical” surfaceM ̃^2 of genusg(so-
lutions of (10.15)) gluing of Hopf tori.

Conclusion

An appearance of topological invariants like the Hopf invariant in such dif-
ferent problems of mathematics, physics, and even biology is far from being
occasional. It reflects that the background of many modern constructions is
based on the common topological ideas.
We have no opportunity to deeper in the relevant theories. We shall give
only a short list of the topics with some references where a reader will be able
to find both applications and generalizations of Hopf theory and a theme for
the investigations:

1. The theory of multivalued functionals [14, 15] and the chapter by Million-
   schikov in this book.


2. Topological field theory [16, 24].


3. Fractional Statistics and quantum Hall effect [17, 18].


4. Topological invariants in magnetohydrodynamics [19, 20].


5. Knotlike configurations in relativistic field theory [21].


6. Quantum computations [25].

References

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6. J. White, Geometry and topology of DNA–protein interactions, inNew Scientific
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10. E.I. Kats, M.I. Monastyrsky, JETP, 91 , 1279–1285 (2000)