2011/12/31 by Christopher Tuffley
Mathematics · #math.GT #msc:57Q45 #msc:57M15 #msc:57Q35
paper · pdf · doi:10.2140/agt.2013.13.1579
published as Algebr. Geom. Topol. 13 (2013) 1579-1612 · 26 pages, 4 figures. v3: references added, some typos corrected, order of Thms 1.4 and 1.5 reversed, other minor changes in response to referee's comments. v2: added reference to arXiv:0705.2026 and updated abstract and introduction in view of that paper; improved bound in Thm 1.4 from O(p^4) to O(p^2); some additional discussion of results; typos corrected
arxiv created 2012/12/20 · arxiv updated 2014/09/30
Define the complete n-complex on N vertices to be the n-skeleton of an (N-1)-simplex. We show that embeddings of sufficiently large complete n-complexes in R2n+1 necessarily exhibit complicated linking behaviour, thereby extending known results on embeddings of large complete graphs in R3 (the case n=1) to higher dimensions. In particular, we prove the existence of links of the following types: r-component links, with the linking pattern of a chain, necklace or keyring; 2-component links with linking number at least lambda in absolute value; and 2-component links with linking number a non-zero multiple of a given integer q. For fixed n the number of vertices required for each of our results grows at most polynomially with respect to the parameter r, lambda or q.