Every knot, K, in S3 has
associated to it an equivalence class of matrices based on S-equivalence of Seifert
matrices. When the knot is altered by changing a crossing, the S-equivalence class of
the new knot is related to that of the original knot in a very specific way. This
change in the Seifert matrices can be studied without regard to the underlying
geometric situation, leading to a theory of algebraic crossing changes. Thus,
the algebraic unknotting number may be defined as the smallest number
of these algebraic crossing changes necessary to convert a Seifert matrix
for the knot into a matrix for the unknot. A straightforward test of some
well-known knot invariants will reveal that the algebraic unknotting number is
one.
