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The palette numbers of torus knots

Department of Mathematics, Kobe University, Rokkodai-cho 1-1, Nada-ku, Kobe 657-8501, Japan

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

E-mail Address: n-takuji@osakac.ac.jp

Corresponding author.

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Yasutaka Nakanishi

Department of Mathematics, Kobe University, Rokkodai-cho 1-1, Nada-ku, Kobe 657-8501, Japan

E-mail Address: nakanisi@math.kobe-u.ac.jp

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Shin Satoh

Department of Mathematics, Kobe University, Rokkodai-cho 1-1, Nada-ku, Kobe 657-8501, Japan

E-mail Address: shin@math.kobe-u.ac.jp

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https://doi.org/10.1142/S0218216517500602Cited by:0 (Source: Crossref)

Abstract

For an effectively $n$ -colorable knot $K$ , the palette number $C_{n}^{*} (K)$ is the minimum number of distinct colors for all effective $n$ -colorings of $K$ . It is known that $C_{n}^{*} (K) \geq 2 + ⌊ {log}_{2} n ⌋$ for any effectively $n$ -colorable knot $K$ . In this paper, we show that for any odd $n \geq 3$ and effectively $n$ -colorable torus knot $K$ it holds that $C_{n}^{*} (K) = 2 + ⌊ {log}_{2} n ⌋$ namely, any effectively $n$ -colorable torus knot has an effectively $n$ -colored diagram with $2 + ⌊ {log}_{2} n ⌋$ colors.

Keywords:

AMSC: 57M25

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