  We present a new 2-variable generalization of the Jones polynomial that can
be defined through the skein relation of the Jones polynomial. The
well-definedness of this new generalization is proved both algebraically and
diagrammatically as well as via a closed combinatorial formula. This new
invariant is able to distinguish more pairs of non-isotopic links than the
original Jones polynomial such as the Thistlethwaite link from the unlink with
two components.
