  The theoretical construction of a traversable wormhole proposed by Morris and
Thorne maintains complete control over the geometry by assigning both the shape
and redshift functions, thereby leaving open the determination of the
stress-energy tensor. This paper examines the effect of introducing the linear
barotropic equation of state $p_r=\omega\rho$ on the theoretical construction.
If either the energy density or the closely related shape function is known,
then the Einstein field equations do not ordinarily yield a finite redshift
function. If, however, the wormhole admits a one-parameter group of conformal
motions, then both the redshift and shape functions exist provided that $\omega
<-1$. In a cosmological setting, the equation of state $p=\omega\rho$, $\omega
<-1$, is associated with phantom dark energy, which is known to support
traversable wormholes.
