  We call a knot $K$ a complete Alexander neighbor if every possible Alexander
polynomial is realized by a knot one crossing change away from $K$. It is
unknown whether there exists a complete Alexander neighbor with nontrivial
Alexander polynomial. We eliminate infinite families of knots with nontrivial
Alexander polynomial from having this property and discuss possible strategies
for unresolved cases.
  Additionally, we use a condition on determinants of knots one crossing change
away from unknotting number one knots to improve KnotInfo's unknotting number
data on 11 and 12 crossing knots. Lickorish introduced an obstruction to
unknotting number one which proves the same result. However, we show that
Lickorish's obstruction does not subsume the obstruction coming from the
condition on determinants.
