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 $K^1_{\mathrm{top}}$? (e.g., For cohomology $H^1(X,\mathbb{Z})\cong\mathbb{Z}^{2g}$ we may take the 1-cochains ``around the holes'')

Furthermore, we define the Mukai vector of $\kappa\in K^1_{top}(X)$ to be $v(\kappa)=\mathrm{ch}(\kappa)\sqrt{\mathrm{td}(X)}$, and the Euler pairing on $K^*_{top}(X)$ by $\langle a,b\rangle=(v(a^\vee),v(b))$ where $(-,-)$ is the pairing in $H^*(X,\mathbb{Q})$. Do we know the pairing with respect to the basis? (The definition of $\mathrm{ch}\colon K^*(X)\to \oplus H^*(X)$ is given in https://www.maths.ed.ac.uk/~v1ranick/papers/ahvbh.pdf)