## Abstract

A local move is a pair of tangles with same end points. Habiro defined a system of local moves, C_{n}-moves, and showed that two knots have the same Vassiliev invariants of order ≤ n - 1 if and only if they are transformed into each other by C_{n}-moves. We define a local move, β_{n}-move, which is obtained from a C_{n}-move by duplicating a single pair of arcs with same end points. Then we immediately have that a C_{n}+1-move is realized by a β_{n}-move and that a β_{n},-move is realized by twice C_{n}-moves. In this note we study the relation between C_{n}-move and β _{n}-move, and in particular, give answers to the following questions: (1) Is a β_{n}-move realized by a finite sequence of C _{n}+1-moves? (2) Is C_{n}-move realized by a finite sequence of β_{n},-moves?

Original language | English |
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Pages (from-to) | 839-851 |

Number of pages | 13 |

Journal | Journal of Knot Theory and its Ramifications |

Volume | 15 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2006 Sep |

Externally published | Yes |

## Keywords

- C-move
- Local move
- Δ-move
- β-move

## ASJC Scopus subject areas

- Algebra and Number Theory