ИСТИНА

An algebraic link is the intersection of a germ of a plane analytic curve $(C,0)\subset(\CC^2,0)$ (reducible or irreducible) with the sphere $S^3_{\varepsilon}$ of a small radius $\varepsilon$ centred at the origin. The number of components of the link is equal to the number of components of the curve germ. Thus, if the curve $(C,0)$ is irreducible (consists of one component), the corresponding algebraic link is a knot. For an irreducible curve germ $(C,0)$ one can associate the following invariant (in principle, an analytic one). Let $\varphi:(\CC,0)\to (C,0)$ be a parametrization (an uniformization) of the curve $(C,0)$. For a germ $f\in\calO_{\CC^2,0}$ of a function in two variables, let $v(f)$ be the degree of the leading term in the power series decomposition $f\circ \varphi(\tau)=a\tau^{v(f)}+ terms\ of\ higher\ degree$. If $f\circ\varphi\equiv 0$, $v(f):=+\infty$. ($v$ is a valuation on the ring $\calO_{\CC^2,0}$.) For $k\in\ZZ$, let $J(k)=\{f\in\calO_{\CC^2,0}: v(f)\ge k\}$. The Poincar\'e series of the filtration $\{J(v)\}$ is $P_C(t)=\sum_{k=0}^{\infty}\dim J(k)/J(k+1) t^k$. It appears that the Poincar\'e series $P_C(t)$ coincides with the Alexander polynomial of the knot $(C,0)\cap S^3_{\varepsilon}\subset S^3_{\varepsilon}$ divided by $1-t$. (Up to now the statement has no conceptual proof: it is obtained by a direct computation of the Alexander polynomial and of the Poincar\'e series in terms of a resolution of the curve $(C,0)$ and comparison of the results.) There is a generalisation of the notion of the Poincar\'e series and of the statement about its coincidence with the Alexander polynomial for reducible curves. Let $\CC^2$ be the complexification of the real plane $\RR^2$. For a curve germ $(C,0)\subset (\CC^2,0)$, one can consider the fitration defined in the same way on the ring $\calE_{\RR^2,0}$ of real analytic functions in two variables and define (some) analogues of the Poincar\'e series. They appear to be closely related (one of them coincides) with the Poincar\'e series of another curve. This leads to the problem to define an analogue of the Alexander polynomial for a link in the 3-sphere with the real structure, i.e. with an involution whose fixed point set is the trivial (non-linked) knot.