[Math] First homology group of a double torus (genus 2 surface) – intuition

Question

First homology group of a double torus is $H_1(T^2\#T^2)=\mathbb Z^4,$ (where # stands for a connected sum) which – for me – intuitively means there are 4 different cycles up to homotopy, the black ones. But what about those two yellow? Are they some kind of combination of the four?

Answer 1

Here's some pictures that show how to get the big yellow loop from the two horizontal black loops and the smaller yellow loop from the two vertical black loops.

Big yellow:

Small yellow: