  The phase diagram of lowest-energy vortices in the polar phase of spin-1
Bose--Einstein condensates is investigated theoretically. Singly quantized
vortices are categorized by the local ordered state in the vortex core and
three types of vortices are found as lowest-energy vortices, which are elliptic
AF-core vortices, axisymmetric F-core vortices, and N-core vortices. These
vortices are named after the local ordered state, ferromagnetic (F),
antiferromagnetic (AF), broken-axisymmetry (BA), and normal (N) states apart
from the bulk polar (P) state. The N-core vortex is a conventional vortex, in
the core of which the superfluid order parameter vanishes. The other two types
of vortices are stabilized when the quadratic Zeeman energy is smaller than a
critical value. The axisymmetric F-core vortex is the lowest-energy vortex for
ferromagnetic interaction, and it has an F core surrounded by a BA skin that
forms a ferromagnetic-spin texture, as exemplified by the localized Mermin--Ho
texture. The elliptic AF-core vortex is stabilized for antiferromagnetic
interaction; the vortex core has both nematic-spin and ferromagnetic orders
locally and is composed of the AF-core soliton spanned between two BA edges.
The phase transition from the N-core vortex to the other two vortices is
continuous, whereas that between the AF-core and F-core vortices is
discontinuous. The critical point of the continuous vortex-core transition is
computed by the perturbation analysis of the Bogoliubov theory and the
Ginzburg--Landau formalism describes the critical behavior. The influence of
trapping potential on the core structure is also investigated.
