On algebraic unknotting numbers of knots

We show that the algebraic unknotting number of a classical knot $K$, defined by Murakami [9], is equalto the minimum number of unknotting operations necessary to transform K to a knot with trivial Alexander polynomial. Furthermore, we define a new operation, called an elementary twisting operation, for smooth (2n−1)-knots with n≥1 and odd, and show that this is an unknotting operation for simple (2n−1)-knots. Moreover, the unknotting number of a simple (2n−1)-knot defined by using the elementary twisting operation isequal to the algebraic unknotting number of the S-equivalence class of its Seifert matrix ifn≥3 .