Fibred knots and disks with clasps
If K is a null-homotopic knot in a closed, orientable 3-manifold M, then K bounds a disk-with-clasps and is therefore null-homotopic in a handlebody in M. Using the fact that M contains a fibred knot, it is shown that if every knot in M is null-homotopic in a handlebody of genus 2 then M is homeomorphic to \(S^ 3\). This generalizes results of \textit{R. H. Bing} [Ann. Math., II. Ser. 68, 17-37 (1958; Zbl 0081.392)] and \textit{D. R. McMillan} jun. [Trans. Am. Math. Soc. 98, 350-367 (1961; Zbl 0098.147)] where the same conclusion is obtained with the handlebody of genus 2 replaced by a 3-ball and solid torus respectively.
null-homotopic knot in a closed, orientable 3-manifold
null-homotopic in a handlebody of genus 2
scientific article; zbMATH DE number 3925806