Topology and stability of a (2+1)-dimensional black hole in f(Q) gravity

In this work, we construct a novel exact solution in the framework of symmetric teleparallel gravity (STG), specifically in the context of three-dimensional $f\left(Q\right)$ gravity, where $Q$ is the non-metricity scalar. Utilizing a spherically symmetric $(2+1)$-dimensional metric and the coincident gauge condition, we derive a new class of black hole solutions characterized by an analytic form of $f\left(Q\right)$ that generalizes the Banados-Teitelboim-Zanelli (BTZ) solution. The resulting solution generalizes the BTZ black hole by incorporating a dimensional deformation parameter $a_1$, yielding deviations from general relativity due to non-metricity corrections. This solution reduces to the standard BTZ geometry in the limit of vanishing deformation parameter $a_1$, while exhibiting distinctive features when $a_1\ne 0$, including deviation in curvature and non-metricity scalars. We conduct a thorough analysis of the thermodynamic properties of the resulting black hole, including its Hawking temperature, entropy, and heat capacity, confirming its thermodynamic stability and demonstrating the validity of the first law of thermodynamics. Furthermore, we explore the topological classification of the black hole using the generalized free energy method, demonstrating the existence of nontrivial topological charges associated with its horizon structure.