Isogeometric analysis (IGA) is a numerical analysis method that provides arbitrary continuity between elements and the high accuracy compared to standard finite element methods. When performing adaptive mesh refinement (AMR) with IGA, we must use analysis-suitable (AS) T-splines as basis functions instead of non-uniform rational B-splines (NURBS) to handle local change in the mesh structure. IGA should be examined over a range of different polynomial degrees, since it increases the computational precision and mitigates numerical locking. However, most previous studies on adaptive IGA are limited to cubic degrees. This limitation is primarily attributed to the definition of T-meshes, which represent the parametric domain for T-splines. The geometric locations of the anchors representing the control points in T-meshes depend on the polynomial degrees. To address this issue, we propose a topological data structure for T-meshes instead of the conventional index-grid-based data structure. This proposed design allows for flexible handling of local T-mesh modifications and polynomial degree variations. Furthermore, we organize the Workflow of mesh refinement into three domains: the parametric domain, element domain, and physical domain. Conventional approaches require frequent transitions between these domains, which tightly couple the computational procedures and make parallelization difficult. With this organization, almost all operations in the element and physical domains can be executed in parallel, leading to efficient and fast mesh generation. We validate the proposed framework using numerical examples combining growth theory with membrane structural problems.