On the Non-Existence of Elements of Hopf Invariant One

Adams, J. F.

doi:10.2307/1970147

Cited by 704 publications

(628 citation statements)

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“…Lin and Y. Yang In order to prove the theorem, we need to recall the following facts concerning the possible values of the Hopf invariant (Bott & Tu 1982;Husemoller 1994 (Adams 1960;Adams & Atiyah 1966;Husemoller 1994). (iii) For any n, there is always a map S 4nK1 / S 2n with the Hopf invariant equal to any even number.…”

Section: Universal Upper Boundmentioning

confidence: 99%

Universal growth law for knot energy of Faddeev type in general dimensions

Lin

Yang²

2008

Proc. R. Soc. A.

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The presence of a fractional-exponent growth law relating knot energy and knot topology is known to be an essential characteristic for the existence of 'ideal' knots. In this paper, we show that the energy infimum E N stratified at the Hopf charge N of the knot energy of the Faddeev type induced from the Hopf fibration S 4nK1 / S 2n (n R1) in general dimensions obeys the sharp fractional-exponent growth law E N wjN j p , where the exponent p is universally rendered as pZ ð4nK1Þ=4n, which is independent of the detailed fine structure of the knot energy but determined completely by the dimensions of the domain and range spaces of the field configuration maps.

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Section: Universal Upper Boundmentioning

confidence: 99%

Universal growth law for knot energy of Faddeev type in general dimensions

Lin

Yang²

2008

Proc. R. Soc. A.

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“…We know from Hurwitz theorem (see reference [24]) that octonions are one of the alternative division algebras (the others are the real numbers, the complex numbers and the quaternions). While among the only parallelizable spheres we find S 7 (the other are the spheres S 1 and S 3 [38]). This distinctive and fundamental role played by the Fano matroid, octonions and S 7 in such different areas in mathematics as combinatorial geometry, algebra and topology respectively lead us to believe that the relationship between these three concepts must have a deep significance not only in mathematics, but also in physics.…”

Section: -Matroid Theory and Supergravitymentioning

confidence: 68%

Matroid theory and Chern–Simons

Nieto

Marín

2000

Journal of Mathematical Physics

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It is shown that matroid theory may provide a natural mathematical framework for a duality symmetries not only for quantum Yang-Mills physics, but also for M-theory. Our discussion is focused in an action consisting purely of the Chern-Simons term, but in principle the main ideas can be applied beyond such an action. In our treatment the theorem due to Thistlethwaite, which gives a relationship between the Tutte polynomial for graphs and Jones polynomial for alternating knots and links, plays a central role. Before addressing this question we briefly mention some important aspects of matroid theory and we point out a connection between the Fano matroid and D=11 supergravity. Our approach also seems to be related to loop solutions of quantum gravity based in Ashtekar formalism.

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“…We very briefly describe the portion of Adams' work on cohomology operations ( [1]) relevant to this proof. All the spaces we will deal with will be approximately k-connected, and the results we state will be correct in a range of dimensions up to about 2k.…”

Section: Proof Of Theorem 54mentioning

confidence: 99%