Ribbon Hopf algebras from group character rings Peter Jarvis School of Physical Sciences University of Tasmania Joint work with Bertfried Fauser and Ronald King We study the diagram alphabet of knot moves associated with the character rings of certain matrix groups. The primary object is the Hopf algebra 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, as well as the formal character rings of algebraic subgroups of $GL(n)$, comprised of matrix transformations leaving invariant a fixed but arbitrary tensor of Young symmetry type $\pi$, which include the orthogonal and symplectic groups as special cases. 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.