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' is obtained from a link L=(K1,…,Km) by replacing each knot Ki with its (pi,qi)-cable for some pi and qi, we have v(L')=(p1⋯ pm)kv(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 S3 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≥ 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 ΩL(x1,…,xn) expanded as a formal power series in Conway's variables zi=xi−xi−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 μ̅-invariants), and applications to links of solenoids.
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BibTeX
@misc{melikhov2026satellites,
title = {{Satellites and invariants of links}},
author = {Sergey A. Melikhov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.33309},
eprint = {2609.33309},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33309},
}
RIS
TY - UNPB
TI - Satellites and invariants of links
AU - Sergey A. Melikhov
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2609.33309
UR - https://arxiv.org/abs/2609.33309
AB - An invariant v of m-component links is called "cableable" if there exists a k such that whenever a link L' is obtained from a link L=(K_1,…,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')=(p_1⋯ p_m)^kv(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≥ 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 Ω_L(x_1,…,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 μ̅-invariants), and applications to links of solenoids.
ER -