  The topological R\'enyi and entanglement entropies depend on the bipartition
of the manifold and the choice of the ground states. However, these
entanglement quantities remain invariant under a coordinate transformation when
the bipartition also undergoes the same transformation. In the context of
topological quantum field theories, these coordinate transformations reduce to
representations of the mapping class group on the manifold of the Hilbert
space. We employ this invariant property of the R\'enyi and entanglement
entropies under coordinate transformations for TQFTs in (2 + 1) dimensions on a
torus with various bipartitions. By utilizing the replica trick and the surgery
method to compute the topological R\'enyi and entanglement entropies, the
invariant property results in Verlinde-like formulas. Furthermore, for the
bipartition with interfaces as two non-intersecting torus knots, an $SL(2,
\mathbb{Z})$ transformation can untwist the torus knots, leading to a simple
bipartition with an effective ground state. This invariant property allows us
to demonstrate that the topological entanglement entropy has a lower bound $-2
\ln D$, where $D$ is the total quantum dimensions of the system.
