On a continuation of quaternionic and octonionic logarithm along curves and the winding number

This paper focuses on the problem of finding a continuous extension of the hypercomplex logarithm along a path. While a branch of the complex logarithm can be defined in a small open neighbourhood of a strictly negative real point, no continuous branch of the hypercomplex logarithm can be defined in any open set $A\subset \mathbb{K}\setminus \{0\}$ which contains a strictly negative real point $x_0$ (here $\mathbb{K}$ represents the algebra of quaternions or octonions). To overcome these difficulties, we introduced the logarithmic manifold ${\mathcal{E}}_{\mathbb{K}}^{+}$ and then showed that if $q\in \mathbb{K},q=x+Iy$ then $E(x+Iy)$ is an immersion and a diffeomorphism between $\mathbb{K}$ and ${\mathcal{E}}_{\mathbb{K}}^{+}$. In this paper, we consider lifts of paths in $\mathbb{K}\setminus \{0\}$ to the logarithmic manifold ${\mathcal{E}}_{\mathbb{K}}^{+}$; even though $\mathbb{K}\setminus \{0\}$ is simply connected, in general, given a path in $\mathbb{K}\setminus \{0\}$, the existence of a lift of this path to ${\mathcal{E}}_{\mathbb{K}}^{+}$ is not guaranteed. There is an obvious equivalence between the problem of lifting a path in $\mathbb{K}\setminus \{0\}$ and the one of finding a continuation of the hypercomplex logarithm ${\log }_{\mathbb{K}}$ along this path.