## Quasipositive links and electromagnetism.(English)Zbl 1484.57002

A link $$L\in S^3$$ is called quasipositive if it is the closure of a braid of the form $$\prod\limits_{j=1}^{l}w_{j}\sigma_{i_j}w_{j}^{-1}$$, where $$w_j$$ is any braid word and $$\sigma_{i_j}$$ is a positive standard generator of the braid group. A link $$L\in S^3$$ is a transverse $$\mathbb{C}$$-link if it is the transverse intersection $$f^{-1}(0)\cap S^3$$ of the vanishing set of a complex polynomial $$f : \mathbb{C}^{2} \rightarrow \mathbb{C}$$ and the unit sphere $$S^3$$. L. Rudolph has shown that every quasipositive link is a transverse $$\mathbb{C}$$-link in [Topology 22, 191–202 (1983; Zbl 0505.57003)]. Conversely, M. Boileau and S. Orevkov have shown that every transverse $$\mathbb{C}$$-link is quasipositive in [C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 9, 825–830 (2001; Zbl 1020.32020)].
In [in: Handbook of knot theory. Amsterdam: Elsevier. 349–427 (2005; Zbl 1097.57012)], L. Rudolph has shown that every link in $$S^3$$ is the sublink of a quasipositive link. In the paper under review, the author constructively proves that every link is the sublink of a quasipositive link $$\tilde{L}$$ which is a satellite of the Hopf link. Suppose $$\tilde{L}=f^{-1}(0)\cap S^3$$ for some complex polynomial $$f : \mathbb{C}^{2} \rightarrow \mathbb{C}$$. The construction gives upper bounds for the degrees of $$f$$.
For every link $$L$$, the author constructs an electromagnetic field $$\mathbf{F}_t=\mathbf{B}_{t}+i\mathbf{E}_{t}:\mathbb{R}^3\rightarrow \mathbb{C}^3$$ that satisfies Maxwell’s equation and such that the set of null lines $$\mathbf{F}^{-1}_t(0,0,0)$$ contains $$L$$ for all times $$t$$.
The author also shows that the time evolution of electromagnetic fields as given by H. Bateman’s construction [The mathematical analysis of electrical and optical wavemotion on the basis of Maxwell’s equations. Cambridge: University Press (1915; JFM 45.1324.05)] and a choice of time-dependent stereographic projection can be understood as a continuous family of contactomorphisms with knotted field lines of the electric and magnetic fields corresponding to Legendrian knots.

### MSC:

 57K10 Knot theory 78A25 Electromagnetic theory (general)
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