I read on Wikipedia that Frankl and Prontrjagin were the first to prove that a link $\mathbb{R}^3$ bounds a surface. A few years later Seifert published a proof using the "Seifert algorithm" which seems to be the proof preferred in text books on knot theory. I was wondering if anyone knows an English reference for the original proof or any information about its flavor. Thank you for your time.


You ask for an English translation of the 1930 paper in German by Frankl and Prontrjagin. There are many such translation requests on MO, so as an experiment, I tried the following: I found a high-quality scan of the paper, OCR'd it and gave it to Google Translate. No editing on my part. Here is the resulting text. Together with the original text for the formulas, it seems workable to me. What do you think?


Here is a picture of the Frankl-Pontrjagin construction, from a paper by Lucien Guillou and Alexis Marin:


The inscription reads: "The Frankl-Pontryagin surface of the trefoil knot is depicted with two 'windows' showing what happens inside."

The picture is accompanied in the paper by an actual exposition of a generalization of the Frankl-Pontrjagin construction. This is section 4 of Guillou and Marin's comments 'On the second paper' by Rokhlin in their book "A la Recherche de la Topologie Perdue" (in French; the scan above is from the Russian translation).

  • $\begingroup$ "Dva okoshka" has more charm than the plain "two windows. :-) $\endgroup$ – Włodzimierz Holsztyński Feb 17 '13 at 0:08
  • 1
    $\begingroup$ I think the picture is broken, sadly making this answer useless :( $\endgroup$ – Danu Dec 15 '16 at 10:47

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