### Abstract

Levine introduced clover links to investigate the indeterminacy of Milnor invariants of links. He proved that for a clover link, Milnor numbers of length up to 2k + 1 are well-defined if those of length ≤ k vanish, and that Milnor numbers of length at least 2k + 2 are not well-defined if those of length k + 1 survive. For a clover link c with vanishing Milnor numbers of length ≤ k, we show that the Milnor number μ_{c}(I) for a sequence I is well-defined by taking modulo the greatest common divisor of the μ_{c}(J)′s, where J is any proper subsequence of I obtained by removing at least k + 1 indices. Moreover, if I is a non-repeated sequence of length 2k + 2, the possible range of μ_{c}(I) is given explicitly. As an application, we give an edge-homotopy classification of 4-clover links.

Original language | English |
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Article number | 1650108 |

Journal | International Journal of Mathematics |

Volume | 27 |

Issue number | 13 |

DOIs | |

Publication status | Published - 2016 Dec 1 |

Externally published | Yes |

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### Keywords

- based links
- clover links
- edge-homotopy
- link-homotopy
- Milnor invariants
- spatial graphs

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*International Journal of Mathematics*,

*27*(13), [1650108]. https://doi.org/10.1142/S0129167X16501081