Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/circles/oim/mmks/works2013/rezimin22.pdf Дата изменения: Wed Dec 11 22:45:57 2013 Дата индексирования: Sun Apr 10 19:52:10 2016 Кодировка: Поисковые слова: moon

Rep ort on "A short proof of the Conway-Gordon-Sachs Theorem" by Arseny Zimin The theorem in question says that any PL embedding of K6 , the complete graph on 6 vertices, in 3-dimensional Euclidean space, contains a pair of disjoint linked cycles. The original pro of used the fact that () any two such embeddings are related by isotopy and crossing changes, where edges are allowed to pass through each other. The present pro of pro ceeds by considering a general position pro jection onto a plane of a K5 subgraph of K6 and using the fact that () for any such map the sum of the numbers of intersection points of the images of edges, over all unordered pairs of disjoint edges, is o dd. This but these than the regarded is a nice piece of work for a high scho ol student. There are errors in the pro of can be corrected. I would question the title: the pro of is actually much longer original pro of. On the other hand, it do esn't depend on fact (), so it might be as more elementary.

Lemma 5 is obviously false as stated: the sum should be over all unordered pairs of disjoint edges, i.e. it should be fact (), and this is what is assumed in the pro of. (In the earlier argument, at the top of p.4, this do esn't matter because here, edges e and e that share a vertex automatically have lk (e, e ) = 0.) Even then the pro of of Lemma 5 is not quite correct. In the discussion of Case 2, at the bottom of p.5, the existence of a certain broken line A is asserted. However, the attached figure shows that A may not exist. Of course the pro of could be mo dified to take this into account, for example by inducting on the number of points of intersection of f (e) and f (e). The notation in the final argument, on p.6, is confusing. First, the definition of Cij should be that it is the cycle in K5 on the three vertices other than i and j . Second, the vertex a has been specified (on l.7), so it shouldn't then be used to denote a generic vertex in the subsequent discussion.