Timeline for Uniform closure of a neighbourhood complex in the tritetragonal tiling
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 21, 2020 at 7:54 | comment | added | Hans-Peter Stricker | Let us continue this discussion in chat. | |
| Aug 21, 2020 at 7:38 | comment | added | Hans-Peter Stricker | Because my complex has $v = 36$ vertices. | |
| Aug 21, 2020 at 7:37 | comment | added | M. Winter | @Hans-PeterStricker Why $\chi=-9$? | |
| Aug 21, 2020 at 7:36 | comment | added | Hans-Peter Stricker | $108 = -12\chi$, $\chi = -9$ (I was only talking about my specific complex. For this one, non-existence per se can be proved in principle.) | |
| Aug 21, 2020 at 7:35 | comment | added | M. Winter | @Hans-PeterStricker Where is this bound 108 from? Based on my answer, a large number of edges means that we need a large negative genus. But is not a proof of non-existence per se. | |
| Aug 21, 2020 at 7:32 | comment | added | Hans-Peter Stricker | I counted $e$: Since $64$ edges are already present and $24 \times 3 + 4 \times 4 = 88$ have to be added, a total number of $152$ results, which far exceeds the number of $108$. Is this a strong proof, that my neighourhood complex cannot be uniformly closed? | |
| Aug 21, 2020 at 7:31 | comment | added | M. Winter | @Hans-PeterStricker It does not have to be the (surface of the) double torus, it can potentially be any $n$-holed torus. All I was able to show was that (if embedded) it must lie on a surface of negative Euler characteristic, and this is (in the oriented case) a torus with at laest two holes. You can have unoriented variants too but I do not know their Euler characteristic off the top of my head. | |
| Aug 21, 2020 at 7:24 | comment | added | Hans-Peter Stricker | Note that the complex I want to close (e.g. to a torus or some other surface) has $36$ vertices (necessary condition on the number of vertices fulfilled!) , so the characteristic has to be $-9$, i.e. a $9$-whole torus, right? $f = 63$ and $e=108$? I have to count. | |
| Aug 21, 2020 at 7:16 | comment | added | Hans-Peter Stricker | Here I read that the solid double torus has $\chi = -1$ and the double torus = genus 2 torus = boundary of solid double torus $\chi = -2$. This means that our "torificated" graphs always come as (or on) boundaries of solid double tori? | |
| Aug 20, 2020 at 16:32 | history | answered | M. Winter | CC BY-SA 4.0 |