  Dye and Kauffman defined surface bracket polynomials for virtual links by use
of surface states, and found a relationship between the surface states and the
minimal genus of a surface in which a virtual link diagram is realized. They
and Miyazawa independently defined a multivariable polynomial invariant of
virtual links. This invariant is deeply related to the surface states. In this
paper, we introduce the notion of surface pole bracket polynomials for link
diagrams in closed surfaces, as a generalization of surface bracket polynomials
by Dye and Kauffman. The polynomials induce the invariant of twisted links
defined by the author before as a generalization of Dye, Kauffman and
Miyazawa's polynomial invariant. Furthermore we discuss a relationship between
curves in surface pole states and variables of the polynomial invariant.
