Based on the vector diffraction theory, this paper investigates the focusing properties of linearly polarized Lorentz-Gaussian vortex beams with sinusoidal phase modulation and discusses the focused light intensity under different parameters in detail. Results show that the focus pattern in the horizontal direction at the focal region can be compressed by increasing the relative Gaussian parameter wx. As the relative Lorentz parameter γy increases, the focus pattern will separate in the vertical direction of the optical field. With the topological charge number m increases, a special tunable optical dark trap focusing mode can be obtained. Through changing the sinusoidal modulation parameter n, the regular trilateral, quadrangle, pentagon, and hexagon shapes of the focusing mode can be correspondingly constructed. Besides, on increasing propagation distance z, the focusing mode in the near focusing region gradually extends outside and always exhibits hexagon-shaped patterns, which reflects that this special focusing mode has a good stability. In addition, the optical gradient force distributions and the field intensity distributions in the longitudinal plane are also investigated to illuminate the applications of these alterable focal patterns. Those novel, to the best of our knowledge, findings may be helpful in applications such as optical manipulation, optical focusing, and imaging.