  We study the diagram alphabet of knot moves associated with the character
rings of certain matrix groups. The primary object is the Hopf algebra Char-GL
of characters of the finite dimensional polynomial representations of the
complex group GL(n) in the inductive limit, realised as the ring of symmetric
functions \Lambda(X) on countably many variables X = {x_1,x_2, ...}. Isomorphic
as spaces are the character rings Char-O and Char-Sp of the classical matrix
subgroups of GL(n), the orthogonal and symplectic groups. We also analyse the
formal character rings Char-H_\pi\ of algebraic subgroups of GL(n), comprised
of matrix transformations leaving invariant a fixed but arbitrary tensor of
Young symmetry type \pi, which have been introduced in [5] (these include the
orthogonal and symplectic groups as special cases). The set of tangle diagrams
encoding manipulations of the group and subgroup characters has many elements
deriving from products, coproducts, units and counits as well as different
types of branching operators. From these elements we assemble for each \pi\ a
crossing tangle which satisfies the braid relation and which is nontrivial, in
spite of the commutative and co-commutative setting. We identify structural
elements and verify the axioms to establish that each Char-H_\pi\ ring is a
ribbon Hopf algebra. The corresponding knot invariant operators are rather
weak, giving merely a measure of the writhe.
