Weierstrass polynomials for links (Q1267332)

There is a natural way of identifying links in the 3-space with polynomial covering spaces over the circle due to the author [Pac. J. Math. 81, 399-410 (1979; Zbl 0413.57001)]. By Alexander's theorem any link \(L\) in the 3-space can be constructed by closing a braid around an axis. Choose a circle orthogonal to the axis for the closed braid and with its centre on the axis. Now project the closed braid onto this circle by an orthogonal projection onto the plane of the circle followed by a radial projection from the centre of the circle. By this projection the closed braid, and also the link \(L\), gets structure as a finite covering space over the circle with monodromy defined by the braid; hence it is a polynomial covering space. In particular, any link \(L\) in the 3-space can be defined by a Weierstrass polynomial over the circle. The equivalence relation for covering spaces over the circle is, however, completely different from that for links in the 3-space. The author studies the connections between polynomial covering spaces over the circle and links in the 3-space. In particular, he determines the Weierstrass polynomials for some of the well-known knots and links in the 3-space. For example, regarded as a covering space over the circle, the torus link of type \((n,m)\) is equivalent to the polynomial covering space associated with the Weierstrass polynomial \(P(x,z) =z^n-x^m\), where \(x\) lies in the circle, and \(z\) lies in the complex plane.