Satellites and invariants of links

Abstract

An invariant $v$ of $m$-component links is called "cableable" if there exists a $k$ such that whenever a link $L^{\prime }$ is obtained from a link $L=(K_1,\dots ,K_m)$ by replacing each knot $K_i$ with its $(p_i,q_i)$-cable for some $p_i$ and $q_i$, we have $v(L^{\prime })=(p_1\cdots p_m{)}^{k}v(L)$. The following problem is implicit in a number of papers by P. M. Akhmetiev and originates from the Arnold-Moffatt program for finding topological lower bounds for the energy of a magnetic field: Does there exist a cableable finite type invariant of links in $S^3$ which is not a function of the pairwise linking numbers? According to Akhmetiev, he solved this problem himself, with the desired invariant being defined for $3$-component links as an analytic expression involving a magnetic field modeled on the given link. We offer a different solution, with the desired invariant for $m$-component links, $m\ge 3$, being a function of the coefficients of the Conway polynomial of the link and its sublinks; the function is given explicitly when $m=3$. Moreover, we show that the cables can be replaced by arbitrary satellites. Much of the proof is a study of low degree coefficients of the Conway potential function ${\Omega }_L(x_1,\dots ,x_n)$ expanded as a formal power series in Conway's variables $z_i=x_i-x_i^{-1}$. We also discuss type $n$ invariants which are "cableable up to an invariant of type $n-1$", some cableable invariants which are not of finite type (particularly a certain modification of Milnor's $\bar{\mu }$-invariants), and applications to links of solenoids.

arXiv comment62 pages, 2 figures. To appear in Zap. Nauchn. Sem. POMI (=J. Math. Sci.) v1: About 3 pages of preliminaries migrated from v1-v3 of arXiv:math/0312007 and will be removed from its future versions (starting from v4). v2 and v3: Some clarifications added regarding the related work of P. M. Akhmetiev