On algebraic unknotting numbers of knots (Q1970886)

The author proves that the algebraic unknotting number of a knot \(K\) is equal to the minimum of unknotting operations necessary to transform \(K\) to a knot with trivial Alexander polynomial. Furthermore, for smooth \((2n-1)\)-knots with \(n\geq 1\) and odd, the author defines a new operation called an elementary twisting operation and shows that this is an unknotting operation for simple \((2n-1)\)-knots. Finally, the author proves that the unknotting number of a simple \((2n-1)\)-knot defined by using the elementary twisting operation is equal to the algebraic unknotting number of the \(S\)-equivalence class of its Seifert matrix if \(n\geq 3\).