Timeline for How many non-homeomorphic collections of $N$ circles in $\mathbb{R}^3$ are there?
Current License: CC BY-SA 3.0
13 events
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| Jan 12, 2018 at 0:09 | answer | added | Ian Agol | timeline score: 6 | |
| Jan 11, 2018 at 22:16 | comment | added | HJRW | I'm still totally unclear what the actual question is. Here's a guess. "How many $N$-component links are there in $S^3$ such that each component is an unknot and the absolute value of the linking number of each pair of components is at most 1?" | |
| Jan 11, 2018 at 18:11 | comment | added | Allen Hatcher | This seems to be an interesting question when "circle" is interpreted in the narrow sense of a Euclidean circle lying in a plane. This rules out the Borromean rings for example, by an observation of Freedman and Skora. Then the question is, how many ambient homeomorphism classes of links of $N$ components have representatives that are collections of (disjoint) Euclidean circles? Intuition suggests this number should be finite. | |
| Jan 11, 2018 at 17:07 | comment | added | j.c. | What exactly do you mean by "hooked"? Do you mean that the linking number between two distinct circles is at most one? Also please say more about how you know that your problem can be translated into graph theory. | |
| Jan 11, 2018 at 17:03 | history | edited | j.c. | CC BY-SA 3.0 |
more specific title
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| Jan 11, 2018 at 16:09 | comment | added | Arnaud Mortier | Good point. I was thinking this because the $N=2$ example given by the OP indicates that the Whitehead link is not regarded as different from the unlink. But the Borromeans raise a different question since they cannot be unlinked by allowing self-intersections. | |
| Jan 11, 2018 at 16:01 | comment | added | Igor Rivin | @ArnaudMortier Not exactly - it depends on the notion of equivalence. For example, are borromean rings the same as three unliked circles or not? | |
| Jan 11, 2018 at 14:43 | comment | added | Arnaud Mortier | The fact that you don't seem to care about the actual embeddings in the way a knot theorist would seems to indicate that your question amounts to finding the number of simple graphs on $N$ vertices. Is it not so? | |
| S Jan 11, 2018 at 14:39 | history | edited | YCor | CC BY-SA 3.0 |
latex and tags
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| S Jan 11, 2018 at 14:39 | history | suggested | Arnaud Mortier | CC BY-SA 3.0 |
latex and tags
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| Jan 11, 2018 at 14:35 | review | First posts | |||
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| Jan 11, 2018 at 14:34 | review | Suggested edits | |||
| S Jan 11, 2018 at 14:39 | |||||
| Jan 11, 2018 at 14:31 | history | asked | Jan | CC BY-SA 3.0 |