  In 1979, Hartley and Kawauchi proved that the Conway polynomial of a strongly
negative amphichiral knot factors as $f(z)f(-z)$. In this paper, we normalize
the factor $f(z)$ to define the half-Conway polynomial. First, we prove that
the half-Conway polynomial satisfies an equivariant skein relation, giving the
first feasible computational method, which we use to compute the half-Conway
polynomial for knots with 12 or fewer crossings. This skein relation also leads
to a diagrammatic interpretation of the degree-one coefficient, from which we
obtain a lower bound on the equivariant unknotting number. Second, we
completely characterize polynomials arising as half-Conway polynomials of knots
in $S^3$, answering a problem of Hartley-Kawauchi. As a special case, we
construct the first examples of non-slice strongly negative amphichiral knots
with determinant one, answering a question of Manolescu. The double branched
covers of these knots provide potentially non-trivial torsion elements in the
homology cobordism group.
