Illinois Journal of Mathematics

On 2-knots with total width eight

Osamu Saeki and Yasushi Takeda

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A $2$-knot is (the isotopy class of) a $2$-sphere smoothly embedded in $4$-space. The apparent contour of a generic planar projection of a $2$-knot divides the plane into several regions, and to each such region, we associate the number of sheets covering it. The total width of a $2$-knot is defined to be the minimum of the sum of these numbers, where we take the minimum among all generic planar projections of the given $2$-knot. In this paper, we show that a $2$-knot has total width eight if and only if it is an $n$-twist spun $2$-bridge knot for some $n \neq\pm1$.

Article information

Illinois J. Math., Volume 52, Number 3 (2008), 825-838.

First available in Project Euclid: 1 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25}
Secondary: 57R45: Singularities of differentiable mappings


Saeki, Osamu; Takeda, Yasushi. On 2-knots with total width eight. Illinois J. Math. 52 (2008), no. 3, 825--838. doi:10.1215/ijm/1254403717.

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