Knotting trivial knots and resulting knot types
Let \((V,K)\) be a pattern (i.e. \(V\) is a standardly embedded solid torus in oriented \(S^ 3\) and \(K\) a knot in \(V\)) and \(f\) an orientation preserving embedding from \(V\) into \(S^ 3\) such that \(f(V)\) is knotted. Answers to the following questions are given depending upon whether the winding number of \(K_ 2\) in \(V\) is zero or not. (1) Suppose that \(K_ 1\) is unknotted and \(K_ 2\) is knotted in \(S^ 3\). Can \(f(K_ 1)\) be ambient isotopic to \(f(K_ 2)\) in \(S^ 3\) for some embedding \(f: V\hookrightarrow S^ 3\)? (2) Suppose that \(K_ 1\) and \(K_ 2\) are both unknotted in \(S^ 3\). How are \((V,K_ 1)\) and \((V,K_ 2)\) related if \(f(K_ 1)\) is ambient isotopic to \(f(K_ 2)\) in \(S^ 3\) for some embedding \(f: V\hookrightarrow S^ 3\)?
scientific article; zbMATH DE number 540671