On the distance between homotopy classes in $W^{1\mathrm{/}p,p}({\mathbb{S}}^1;{\mathbb{S}}^1)$

For every $p\in (1,\mathrm{\infty })$ there is a natural notion of topological degree for maps in $W^{1\mathrm{/}p,p}({\mathbb{S}}^1;{\mathbb{S}}^1)$ which allows us to write that space as a disjoint union of classes, $W^{1/p,p}({\mathbb S}^1;{\mathbb S}^1)=\bigcup_{d\in{\mathbb Z}}$\mathcal{E}$d$\mathrm{.}Foreverypair$d_1,d_2\in {\mathbb Z}$,weshowthatthedistance$$\text{Dist}${W^{1/p,p}}({\mathcal E}{d_1}, {\mathcal E}{d_2}):=\sup_{f\in{\mathcal E}{d_1}}
\inf{g\in{\mathcal E}{d_2}}\ d{W^{1/p,p}}(f, g)$equalstheminimal$W^{1/p,p}$-energyin$$\mathcal{E}${d_1-d_2}$\mathrm{.}Inthespecialcase$p=2$ we deduce from the latter formula an explicit value: $$\text{Dist}${W^{1/2,2}}({\mathcal E}{d_1}, {\mathcal E}{d_2})=2\pi|d_2-d_1|^{1/2}$.