Research ArticleNo Access

Amphicheirality of ribbon $2$ -knots

Tomoyuki Yasuda

Department of Liberal Studies, National Institute of Technology, Nara College, Yamatokoriyama, Nara 639-1080, Japan

E-mail Address: yasuda@libe.nara-k.ac.jp

https://doi.org/10.1142/S0218216520500698Cited by:0 (Source: Crossref)

Abstract

For any classical knot $k^{1}$ , we can construct a ribbon $2$ -knot $spun (k^{1})$ by spinning an arc removed a small segment from $k^{1}$ about $R^{2}$ in $R^{4}$ . A ribbon $2$ -knot is an embedded $2$ -sphere in $R^{4}$ . If $k^{1}$ has an $n$ -crossing presentation, by spinning this, we can naturally construct a ribbon presentation with $n$ ribbon crossings for $spun (k^{1})$ . Thus, we can define naturally a notion on ribbon $2$ -knots corresponding to the crossing number on classical knots. It is called the ribbon crossing number. On classical knots, it was a long-standing conjecture that any odd crossing classical knot is not amphicheiral. In this paper, we show that for any odd integer $n$ there exists an amphicheiral ribbon $2$ -knot with the ribbon crossing number $n$ .

Keywords:

AMSC: 57Q45, 57M25

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