  In this article, we separate the vector and axialvector components of the
tensor diquark operators explicitly, construct the axialvector-axialvector type
and vector-vector type scalar tetraquark currents and scalar-tensor type tensor
tetraquark current to study the scalar, vector and axialvector tetraquark
states with the QCD sum rules in a consistent way. The present calculations do
not favor assigning the $Z_c(4100)$ to be a scalar or vector tetraquark state.
If the $Z_c(4100)$ is a scalar tetraquark state without mixing effects, it
should have a mass about $3.9\,\rm{GeV}$ or $4.0\,\rm{GeV}$ rather than
$4.1\,\rm{GeV}$; on the other hand, if the $Z_c(4100)$ is a vector tetraquark
state, it should have a mass about $4.2\,\rm{GeV}$ rather than $4.1\,\rm{GeV}$.
However, if we introduce mixing, a mixing scalar tetraquark state can have a
mass about $4.1\,\rm{GeV}$. As a byproduct, we obtain an axialvector tetraquark
candidate for the $Z_c(4020)$.
