Knot singularities of harmonic morphisms (Q2716978)

A continuous mapping \(\phi :M \rightarrow N\) of Riemannian manifolds is called a harmonic morphism if for every point \(x \in M\) and every harmonic function \(f\) defined on a neighbourhood of \(\phi (x)\), the composition \(F \circ \phi\) is harmonic on neighbourhood of \(x\). The author deals with aultivalued harmonic morphisms from \(\mathbb{R}^3\) to \(\mathbb{C}= \mathbb{R}^2\) and investigates their singular (branching) sets \(K\). He proves that if \(\phi : \mathbb{R}^3 \rightarrow \mathbb{C}\), \(\phi(x_1,x_2,x_3)=z\), is determined implicitly by the formula NEWLINE\[NEWLINE(1-g(z)^2)x_1 +i(1+g(z)^2)x_2 - 2 g(z)x_3 = 2 h(z),NEWLINE\]NEWLINE where \(g(z)=z^p\), \(h(z)=z^{p+2}+i \beta z^p\), \(\beta \in \mathbb{R}_+\), and \(p=3,5,7, \dots \), then \(K\) has a compact component being a knotted curve. Moreover, the knots obtained for different \(p\) are not isotopic.