  We present an enhanced prime decomposition theorem for knots that gives the
isotopy classes of composite knots that can be constructed from a given list of
prime factors (allowing for the mirroring and orientation reversing for each
factor). Underlying the theorem is an algebraic construction that also allows
for the computation of the intrinsic symmetries (invertibility, chirality,
etc.) of a composite knot from those of the prime factors. We then use this
construction to give a table of composite knots through 12 crossings that can
be constructed from prime factors through 9 crossings. This is more difficult
than it might sound because we must take knot symmetries into account when
generating the table (the square knot and the granny knot are different, though
both are connect sums of two trefoils). The completeness of this table depends
on the conjectural additivity of crossing number under the connected sum.
