A note on the sheet numbers of twist-spun knots. (English) Zbl 1198.57016

Satoh, Shin

Hiroshima Math. J. 40, No. 1, 1-15 (2010).

A diagram of a 2-knot \(K:S^2\to\mathbb{R}^4\) is the image of \(K\) under a generic linear projection to \(\mathbb{R}^3\). It is a union of compact connected surfaces embedded in \(\mathbb{R}^3\). The sheet number \(Sh(K)\) is the minimal number of such surfaces needed for any diagram of \(K\). Here quandle cohomology is used to provide lower bounds for \(Sh(K)\). If \(K\) is an \(r\)-twist spin of a 1-knot \(k\) which has a minimal diagram with a clasp then \(Sh(K)\) is bounded above by \((2c(k)-5)r+2\), where \(c(k)\) is the crossing number of \(k\). Together these suffice to determine \(Sh(K)\) for \(K\) the 2- or 3-twist spin of the trefoil knot.

Reviewer: Jonathan A. Hillman (Sydney)

MSC:

57Q45

Knots and links in high dimensions (PL-topology) (MSC2010)

Keywords:

colouring; crossing number; diagram; quandle; Reidemeister move; sheet number; triple point