Cn-move and its duplicated move of links

Kazuaki Kobayashi; Tetsuo Shibuya; Akira Yasuhara

doi:10.1142/S0218216506004774

C_n-move and its duplicated move of links

Kazuaki Kobayashi, Tetsuo Shibuya, Akira Yasuhara

Research output: Contribution to journal › Article › peer-review

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
Pages (from-to)	839-851
Number of pages	13
Journal	Journal of Knot Theory and its Ramifications
Volume	15
Issue number	7
DOIs	https://doi.org/10.1142/S0218216506004774
Publication status	Published - 2006 Sep 1
Externally published	Yes

Keywords

C-move
Local move
Δ-move
β-move

ASJC Scopus subject areas

Algebra and Number Theory

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title = "Cn-move and its duplicated move of links",

abstract = "A local move is a pair of tangles with same end points. Habiro defined a system of local moves, Cn-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 Cn-moves. We define a local move, βn-move, which is obtained from a Cn-move by duplicating a single pair of arcs with same end points. Then we immediately have that a Cn+1-move is realized by a βn-move and that a βn,-move is realized by twice Cn-moves. In this note we study the relation between Cn-move and β n-move, and in particular, give answers to the following questions: (1) Is a βn-move realized by a finite sequence of C n+1-moves? (2) Is Cn-move realized by a finite sequence of βn,-moves?",

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AU - Kobayashi, Kazuaki

AU - Shibuya, Tetsuo

AU - Yasuhara, Akira

PY - 2006/9/1

Y1 - 2006/9/1

N2 - A local move is a pair of tangles with same end points. Habiro defined a system of local moves, Cn-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 Cn-moves. We define a local move, βn-move, which is obtained from a Cn-move by duplicating a single pair of arcs with same end points. Then we immediately have that a Cn+1-move is realized by a βn-move and that a βn,-move is realized by twice Cn-moves. In this note we study the relation between Cn-move and β n-move, and in particular, give answers to the following questions: (1) Is a βn-move realized by a finite sequence of C n+1-moves? (2) Is Cn-move realized by a finite sequence of βn,-moves?

AB - A local move is a pair of tangles with same end points. Habiro defined a system of local moves, Cn-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 Cn-moves. We define a local move, βn-move, which is obtained from a Cn-move by duplicating a single pair of arcs with same end points. Then we immediately have that a Cn+1-move is realized by a βn-move and that a βn,-move is realized by twice Cn-moves. In this note we study the relation between Cn-move and β n-move, and in particular, give answers to the following questions: (1) Is a βn-move realized by a finite sequence of C n+1-moves? (2) Is Cn-move realized by a finite sequence of βn,-moves?

KW - C-move

KW - Local move

KW - Δ-move

KW - β-move

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