Abstract:
An algebraic link is the intersection of a germ of a plane analytic curve $(C,0)\subset(\mathbb C^2,0)$ (reducible or irreducible) with the sphere $S^3_{\varepsilon}$ of a small radius $\varepsilon$ centred at the origin. To an algebraic link one associates an analytic invariant: the so called Poincaré series. For an irreducible curve germ it is defined in the following way. Let $\varphi:(\mathbb C,0)\to (C,0)$ be a parametrization (an uniformization) of the curve $(C,0)$. For a germ $f\in\mathcal O_{\mathbb C^2,0}$ of a function in two variables, let $v(f)$ be the degree of the leading term in the Taylor decomposition $f\circ \varphi(\tau)=a\tau^{v(f)}+ \text{terms of higher degree}$. If $f\circ\varphi\equiv 0$, $v(f):=+\infty$. ($v$ is a valuation on the ring $\mathcal O_{\mathbb C^2,0}$.) For $k\in\mathbb Z$, let $J(k)=\{f\in\mathcal O_{\mathbb C^2,0}: v(f)\ge k\}$. The Poincaré series is $P_C(t)=\sum_{k=0}^{\infty}\dim (J(k)/J(k+1))\cdot t^k$. It appears that the Poincaré series $P_C(t)$ coincides with the Alexander polynomial of the link (divided by $(1-t)$ for a knot). The Alexander polynomial, and thus the Poincaré series, determines the topology of the curve (and therefore of the algebraic link).
Assume that the complex plane has a fixed structure as the complexification of the real one. One can consider algebraic links in the three-sphere $S^3_{\varepsilon}$ with this additional structure. In this setting, one can consider an analogue of the Poincaré series defined by the filtration on the ring of real functions. For knots, this Poincaré series determines the equisingularity type of the knot in the sence of Zariski. It is not clear whether it determines the “real” topology of the knot in a natural sense.
