  This paper is concerned with ground states of attractive Bose gases confined
in an anharmonic trap $V(x)=\omega(|x|^2+k|x|^4)$ rotating at the velocity
$\Omega>0$, where $\omega>0$ denotes the trapping frequency, and $k>0$
represents the strength of the quartic term. It is known that for any
$\Omega>0$, ground states exist in such traps if and only if $0<a<a^*$, where
$a^*:=\|Q\|^{2}_{2}$ and $Q>0$ is the unique positive solution of $\Delta
Q-Q+Q^{3}=0$ in $\mathbb{R}^2$. By analyzing the refined energies and
expansions of ground states, we prove that there exists a constant $C>0$,
independent of $0<a<a^*$, such that ground states do not have any vortex in the
region $R(a):=\big\{x\in\mathbb{R}^2:\, |x|\leq
C(a^*-a)^{-\frac{1-6\beta}{20}}\big\}$ as $a\nearrow a^*$, for the case where
$\omega=\frac{3\Omega^2}{4}$, $k=\frac{1}{6}$, and $\Omega=C_0(a^*-a)^{-\beta}$
varies for some $\beta\in [0,\frac{1}{6})$ and $C_0>0$.
