Quasipositive links and electromagnetism

Abstract

For every link L we construct a complex algebraic plane curve that intersects S 3 transversally in a link L that contains L as a sublink. This construction proves that every link L is the sublink of a quasipositive link that is a satellite of the Hopf link. The explicit construction of the complex plane curve can be used to give upper bounds on its degree and to create arbitrarily knotted null lines in electromagnetic fields, sometimes referred to as vortex knots. Furthermore, these null lines are topologically stable for all time. We also show that the time evolution of electromagnetic fields as given by Bateman's construction 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.

Knotted surfaces as vanishing sets of polynomials

“…with the data points We now perform two trigonometric interpolations, one for each component. For each i = 1, 2 we interpolate the data points − 0.371 cos(3t) − 0.076 cos(4t) − 0.100 cos(5t), (8) shown in Figure 1g). All coefficients are rounded to 3 significant digits.…”

“…In particular, the polynomial f λ is obtained from the braid polynomial by substituting v for e it and v for e −it . In [8] the first author presented an alternative construction of polynomials for (classical) links. Substituting v for e it and 1 v for e −it in the polynomial expression for g λ yields a rational map whose denominator is some power of v and whose numerator is a holomorphic polynomial in complex variables u and v. The numerator's vanishing set intersects S 3 transversally in a link that contains the desired link as a sublink.…”