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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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References:

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