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RIBBON 2-KNOTS WITH DISTINCT RIBBON TYPES

TOMOYUKI YASUDA

Department of Mathematics, Nara National College of Technology, Yata-cho 22, Yamatokoriyama, Nara 639-1080, Japan

https://doi.org/10.1142/S0218216509007579Cited by:6 (Source: Crossref)

Abstract

A ribbon 2-knot is a 2-sphere in R⁴, which is obtained from m 2-spheres in R⁴ by connecting them with m - 1 pipes. Even in the case that a ribbon 2-knot K² is constructed by two 2-spheres and one pipe, a way of constructing K² is not always unique. That is, K² might have presentations of two or more types.

The presentation that a pipe crosses two spheres n times is said to be an n-crossing ribbon presentation, and also n be the ribbon crossing number of this presentation.

In this note, we will show that for any positive integer i, there exists a ribbon 2-knot with presentations of two distinct types such that the difference of their ribbon crossing number is i; and in addition we will also show that for any positive integer j, there exists a ribbon 2-knot with presentations of j, distinct types such that they have the same ribbon crossing number.

Keywords:

AMSC: 57Q45, 57M25

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