A note on untwisted deform-spun 2-knots (Q1201484)

In Trans. Am. Math. Soc. 250, 311-331 (1979; Zbl 0413.57015), \textit{R. A. Litherland} introduced the process of deform-spinning of which twist- spinning, roll-spinning are particular examples. Given a 1-knot \((S^ 3,K)\), let \(g\) be a self-homeomorphism of \((S^ 3,K)\) with \(g=id\) on a tubular neighbourhood \(K\times D^ 2\) of \(K\). The deform-spun 2-knot corresponding to \(g\) is defined as follows. Fix a point \(z\) on \(K\). Take a ball neighbourhood \(K_ -\) of \(z\) in \(K\), and set \(B_ -=K_ -\times D^ 2\). Let \((B_ +,K_ +)\) be the complementary ball pair of \((B_ - ,K_ -)\) which is the standard ball pair. Then we construct \(\partial(B_ +,K_ +)\times B^ 2\cup_ \partial(B_ +,K_ +)\times_ g\partial B^ 2\), where \[ (B_ +,K_ +)\times_ g\partial B^ 2=(B_ +,K_ +)\times I/((x,0)\sim (g(x),1)\text{ for all }x\in B_ +). \] This is a locally-flat sphere pair depending only on the isotopy class \(\gamma\) of \(g\) \((\text{rel }K\times D^ 2)\). We denote this 2-knot by \((S^ 4,\gamma K)\), and call it the deform-spun knot of \(K\) corresponding to \(\gamma\), or \(g\). In this note we prove: Theorem. There exist infinitely many 1-knots \(K\) and untwisted deformations \(\gamma\) of \(K\) such that the corresponding untwisted deform-spun 2-knots \(\gamma K\) are unknotted.