We report Dirac monopoles with polar-core vortex induced by spin-orbit coupling in ferromagnetic Bose-Einstein condensates, which are attached to two nodal vortex lines along the vertical axis. These monopoles are more stable in the time scale of experiment and can be detected through directly imaging vortex lines. When the strength of spin-orbit coupling increases, Dirac monopoles with vortex can be transformed into those with square lattice. In the presence of spin-orbit coupling, increasing the strength of interaction can induce a cyclic phase transition from Dirac monopoles with polar-core vortex to those with Mermin-Ho vortex. The spin-orbit coupled Bose-Einstein condensates not only provide a new unique platform for investigating exotic monopoles and relevant phase transitions, but also can preserve stable monopoles after a quadrupole field is turned off.
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