Altermagnets (AMs) are an emergent class of magnetic materials that combine properties of ferromagnets and antiferromagnets, exhibiting spin-polarized Fermi surfaces and zero net magnetic moment due to combined time-reversal and crystal symmetry. Here, we construct a Kondo-lattice model on a two-dimensional (2D) square Lieb lattice to investigate the topological properties of AMs. We identify a type-II quantum spin Hall state characterized by spin-polarized counterpropagating edge states. Breaking the $C_{4z}\mathcal{T}$ symmetry, which connects magnetic sublattices, induces a transition to a quantum anomalous Hall state. We further establish a strain-induced mechanism to control these topological phase transitions and present the corresponding phase diagram. Finally, we demonstrate the predicted transitions in monolayer CrO, a realistic altermagnetic candidate, using first-principles calculations. Our findings highlight the potential of 2D AMs as a versatile platform for topological spintronics, enabling strain-tunable helical and chiral edge states within a single system.

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• Topological Hall effect
• Topological phase transition
• Altermagnets