How to construct a genus 2 surface from 8-gon?

algebraic-topology reference-request

I am requesting some help or reference for visualization?

I am having a hard time constructing a genus 2 surface from 8-gon. May I request for some reference? Here's the construction I used from Hatcher: enter image description here

And here's my doodle.... I think I didn't get the identification of $b$ correctly at step (c), since I believe it $a$ shall go around the tube... enter image description here


Here are two visualizations. ${}{}{}{}{}{}{}{}{}{}$

two ways of making a genus-two surface from an octogon

Here's a third, from Steve D's comment: third way


You can find this too on p. 301 of Hilbert & Cohn-Vossen "Geometry and the Imagination."

enter image description here


On this page about classification of surfaces (unfortunately only in Frnech) you will find a collection of videos showing how to glue n-gons to get genus-g surfaces

http://analysis-situs.math.cnrs.fr/Classification-des-surfaces-triangulees-par-reduction-a-une-forme-normale.html

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