  This paper presents a new algorithm "A" for constructing Seifert surfaces
from n-bridge projections of links. The algorithm produces minimal complexity
surfaces for large classes of braids and alternating links. In addition, we
consider a family of knots for which the canonical genus is strictly greater
than the genus, (g_c(K) > g(K)), and show that A builds surfaces realizing the
knot genus g(K). We also present a generalization of Seifert's algorithm which
may be used to construct surfaces representing arbitrary relative second
homology classes in a link complement.
