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A C_n-MOVE FOR A KNOT AND THE COEFFICIENTS OF THE CONWAY POLYNOMIAL

Department of Mathematics, College of Arts and Sciences, Tokyo Woman's Christian University, 2-6-1, Zempukuji, Suginami-ku, Tokyo, 167-8585, Japan

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and

HARUMI YAMADA

School of Fundamental Science and Engineering, Waseda University, Okubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan

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https://doi.org/10.1142/S0218216508006403Cited by:8 (Source: Crossref)

Abstract

It is shown that two knots can be transformed into each other by C_n-moves if and only if they have the same Vassiliev invariants of order less than n. Consequently, a C_n-move cannot change the Vassiliev invariants of order less than n and may change those of order more than or equal to n. In this paper, we consider the coefficient of the Conway polynomial as a Vassiliev invariant and show that a C_n-move changes the nth coefficient of the Conway polynomial by ±2, or 0. And for the 2mth coefficient (2m > n), it can change by p or p + 1 for any given integer p.

Keywords:

AMSC: 57M25

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