  The correspondence of the braid group on a handlebody of arbitrary genus to
the algebra of Yang-Baxter and extended reflection equation operators is shown.
Representations of the infinite dimensional extended reflection equation
algebra in terms of direct products of quantum algebra generators are derived,
they lead to a representation of this braid group in terms of $R$-matrices.
Restriction to the reflection equation operators only gives the coloured braid
group. The reflection equation operators, describing the effect of handles
attached to a 3-ball, satisfy characteristic equations which give rise to
additional skein relations and thereby invariants of links on handlebodies. The
origin of the skein relations is explained and they are derived from an
adequately adapted handlebody version of the Jones polynomial. Relevance of
these results to the construction of link polynomials on closed 3-manifolds via
Heegard splitting and surgery is indicated.
