# Picture of the generators of fundamental group of genus g surface

It's a standard result that the fundamental group of a genus g surface has presentation

$$\pi_1(\Sigma_g)=\langle a_1,b_1 \ldots a_g,b_g | \prod_i [a_i,b_i] =1 \rangle.$$

Does anyone have a picture of say a genus 3 surface where these generating loops $$a_i,b_i$$ are explicitly drawn onto the surface?

For example I have see pictures like the one in this answer the fundamental group of a Riemann surface with n points removed or https://www.researchgate.net/figure/Fundamental-group-of-a-genus-3-surface_fig1_226810380 but those sets of 6 loops are different and it's not clear to me which one is correct for the above presentation, or which pairs are $$a_1,b_1$$ or $$a_2,b_2$$ or $$a_3,b_3$$.

Thanks in advance for any help.

• That genus 3 picture does, indeed, give your presentation, up to an appropriate bijection $\{a,b,c,d,e,f\} \leftrightarrow \{a_1,b_1,a_2,b_2,a_3,b_3\}$. Feb 27, 2021 at 1:54
• Mayyyybe this picture is a little more standard, but I don't know if the presentation is going to be any easier to read from it. Feb 27, 2021 at 1:58
• Thanks very much Lee, that picture is really helpful. Feb 28, 2021 at 11:01
• I've turned my comments into an answer. Mar 1, 2021 at 0:36

That genus 3 picture does, indeed, give your presentation, up to an appropriate bijection $$\{a,b,c,d,e,f\} \leftrightarrow \{a_1,b_1,a_2,b_2,a_3,b_3\}$$.