On Algebraic Unknotting Numbers of Knots

Abstract

We show that the algebraic unknotting number of a classical knot $K$, defined by Murakami [9], is equal to 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 $n\geq 1$ 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 is equal to the algebraic unknotting number of the $S$-equivalence class of its Seifert matrix if $n\geq 3$.