  S. Nelson, M. Orrison, V. Rivera {\cite{S}} modified Kauffman's construction
of bracket. Their invariant $\Phi^{\beta}_X$ takes value in a finite ring
$Z_2[t]/(1+t+t^3)$. In this paper, the author generalizes this invariant. The
new invariant takes value in a polynomial ring. Furthermore, for a tricolorable
link diagram, the author gives a bracket invariant which gives lower bound on
number of crossings with different (same) colors.
