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On effective 9-colorings for knots

Takuji Nakamura

Department of Engineering Science, Osaka Electro-Communication University, Hatsu-cho 18-8, Neyagawa 572-8530, Japan

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Yasutaka Nakanishi

Department of Mathematics, Kobe University, Rokkodai-cho 1-1, Nada-ku, Kobe 657-0013, Japan

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, and

Shin Satoh

Department of Mathematics, Kobe University, Rokkodai-cho 1-1, Nada-ku, Kobe 657-0013, Japan

Corresponding author.

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https://doi.org/10.1142/S021821651450059XCited by:7 (Source: Crossref)

Abstract

For a 1- or 2-dimensional knot, we give a lower bound log₂ n + 1 of the minimum number of distinct colors for all effective n-colorings. In particular, we prove that any effectively 9-colorable 1- or ribbon 2-knot is presented by a diagram where exactly five colors of nine are assigned to the arcs or sheets.

Keywords:

AMSC: 57M25, 57Q45

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