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C_n-MOVE AND ITS DUPLICATED MOVE OF LINKS

KAZUAKI KOBAYASHI

Department of Mathematics, Tokyo Woman's Christian University, Zempukuji 2-6-1, Suginami, Tokyo 167-8585, Japan

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TETSUO SHIBUYA

Department of Mathematics, Osaka Institute of Technology, Omiya 5-16-1, Asahi, Osaka 535-8585, Japan

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AKIRA YASUHARA

Department of Mathematics, Tokyo Gakugei University, Nukuikita 4-1-1, Koganei, Tokyo 184-8501, Japan

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https://doi.org/10.1142/S0218216506004774Cited by:0 (Source: Crossref)

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?

Keywords:

AMSC: 57M25

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