Timeline for Is there a "knot theory" for graphs?
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8 events
| when toggle format | what | by | license | comment | |
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| Oct 31, 2011 at 20:24 | comment | added | Sergey Melikhov | Sorry for getting a bit too emotional over the codimension. | |
| Oct 31, 2011 at 14:27 | history | edited | Alfredo Hubard | CC BY-SA 3.0 |
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| Oct 30, 2011 at 19:27 | comment | added | Sergey Melikhov | Finally, the Robertson-Seymour-Thomas result about minors is likely to have an analogue for linkless embeddings of $n$-dimensional simplicial complexes in $\Bbb R^{2n+1}$, $n\ne 2$ (see arxiv.org/abs/1103.5457v2), but I'd skeptical about lower codimension, especially codimension two ($K^n$ in $\Bbb R^{n+2}$ for $n>1$) In fact, I haven't seen any results whatsoever on "codimension two Ramsey theory" ($K^n$ in $\Bbb R^{n+2}$) except for the classical case ($n=1$). | |
| Oct 30, 2011 at 19:20 | comment | added | Sergey Melikhov | (con't) In more detail, higher-dim extensions of Milnor's triple invariant detect a Brunnian "Borromean rings" link of three $S^{2k−1}$'s in $\Bbb R^{3k}$, and a higher-dim counterpart of the Sato-Levine invariant (not the original higher-dim Sato-Levine invariant) detects a "Whitehead link" of two $S^{2k−1}$'s in $\Bbb R^{3k}$, $k\ne 3,7$, which has zero linking number. | |
| Oct 30, 2011 at 19:09 | comment | added | Sergey Melikhov | "... as a famous result of Zeeman says". The fact that every graph unknots in $\Bbb R^n$ for $n>3$ is trivial (use general position) and has nothing to do with Zeeman. Zeeman's result is about piecewise-linear unknotting of spheres in codimension $\ge 3$. But spheres easily link, and connected manifolds easily knot in high codimensions. In fact, "your favorite" link invariant used in "Ramsey link" theory (and I've seen papers dealing with the Sato-Levine invariant and Milnor's triple invariant) probably has a higher-dimensional extension (certainly in those two cases). | |
| Oct 30, 2011 at 18:30 | comment | added | Sergey Melikhov | "in all the previous results is very important that you are dealing with codimension two". The Conway-Gordon/Sachs result has NOTHING to do with codimension two: any map of the $n$-skeleton of the $(2n+3)$-simplex in $\Bbb R^{2n+1}$ contains a pair of disjoint linked boundaries of the $(n+1)$-simplex (Lovasz-Schrijver, ams.org/journals/proc/1998-126-05/S0002-9939-98-04244-0 and Taniyama, pjm.berkeley.edu/pjm/2000/194-2/p14.xhtml; a third proof is in Example 4.7 in arxiv.org/abs/math/0612082 and a fourth in Example 4.9 in arxiv.org/abs/1103.5457v2). | |
| Oct 30, 2011 at 2:55 | comment | added | David Eppstein | I'm leaving this as a comment rather than an answer because it's really the same as what Alfredo already said, but for more of what he mentions in his first paragraph, at a nontechnical level, see en.wikipedia.org/wiki/Linkless_embedding | |
| Oct 30, 2011 at 1:28 | history | answered | Alfredo Hubard | CC BY-SA 3.0 |