identification of polygons sides

hbghlyj

Genus g surface
The following polygon with side identifications is homeomorphic to which surface?
size(4cm);
pair A = (1,0);
pair B = (0.5,sqrt(3)/2);
pair C = (-0.5,sqrt(3)/2);
pair D = (-1,0);
pair E = (-0.5,-sqrt(3)/2);
pair F = (0.5,-sqrt(3)/2);
void d(guide g,real[] pos_array){
draw(g);for(real pos:pos_array){draw(g,p=invisible,Arrow(SimpleHead,angle=45,size=4,Draw(black),position=pos));}}
d(D--C,new real[]{0.5});
d(F--A,new real[]{0.5});
d(E--F,new real[]{0.47,0.53});
d(C--B,new real[]{0.47,0.53});
d(D--E,new real[]{0.45,0.5,0.55});
d(B--A,new real[]{0.45,0.5,0.55});

Czhang271828

Hint: first glue any two pairs of opposite edges to obtain a torus with a pole, then the pole part is trivial...

hbghlyj

page 49

[in a group chat]
Could anyone explain why the weird image on the right is a valid triangulation of P.
Also, is there any general way to obtain the triangulation of a polygon with side identifications?

Joshua: Usually when we first learn triangulation its good to label the points after identifying them, and if you do that you can see that the intersection of any two simplices are either empty or a simplex

In fact, idk if it is mentioned in the lecture notes, but if you triangulate that hexagon with side identifications the simplest possible way, you can see its not a valid simplex because the intersection of any two opposite "triangles" doesnt give a simplex (just gives a union of 3 points)

I dont think there is a very obvious way for every polygon, but certainly the more side identifications the more points you have to add.

In fact for this hexagon,  this triangulation is the 2nd(or close to 2nd anyway) simplest way you can even consider triangulating a hexagon for any triangulation, so it doesnt take very long to trial and error and find the right triangulation
You can see its not a valid triangulation***

Czhang271828

Gluing.
mathcha.io/editor/L0qx0UnjTGxs5x0ZDyF8dP9nYcE4N8rnU4BE2gz

hbghlyj

Czhang271828

hbghlyj 发表于 2023-3-2 23:03
本帖最后由 hbghlyj 于 2023-3-2 16:05 编辑
addEventListener("DOMContentLoaded",draw);
function draw ...

mathcha.io/editor/ZMyPMfQeFJWFzdcMP6wvrHKe7zj5UWqwzdho1Qp0G

现在可以修改了，之前是只读

hbghlyj

Fig. 1

Edge pairing of polygon: (a) $E=2$ and (b) $E=3$
From: Graphs and Closed Surfaces Associated with a Pairing of Edges for Regular Polygons

hbghlyj

另一个类似的
Dyck's surface

hbghlyj

Czhang271828

hbghlyj 发表于 2023-6-1 16:17 本帖最后由 hbghlyj 于 2023-6-1 09:22 编辑

光速画了一个, 草稿本随意编辑.

　

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diagram-20230601.svg (14.93 KB, 下载次数: 19)

2023-6-1 16:32 上传

点击文件名下载附件

　

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