Let $X$ be a compact Riemann surface of genus $g$, then $K^1_{\mathrm{top}}(X)\cong\mathbb{Z}^{2g}$. Is there a explicit description of a set of basis of $K^1_{\mathrm{top}}$ and what is the Euler pairing $\langle x,y\rangle=\chi(x,y)$ for the basis? (e.g., For cohomology $H^1(X,\mathbb{Z})\cong\mathbb{Z}^{2g}$ we may take the 1-cochains ``around the holes'')