Ring-valued knot invariants

Let $R$ be a non-trivial ring (i.e. $1\ne 0$). Let $A(R)$ be the group of
invertible upper-triangular two-by-two matrices with elements in $R$ and with
the upper diagonal element equal to one.

Let $G$ be a group and $H$ a subgroup of $G$. Through the study
of the set of equivalence classes of special group homomorphisms from $G$ to
$A_R$ (special means that the subgroup $H$ is represented by diagonal matrices
and all non-zero off-diagonal elements are invertible), one can define a set of
rings associated with the pair $(G,H)$.

Of particular interest is the case, where $G$ is a knot group and $H$ is the
peripheral subgroup of $G$.  For example, the associated rings of the trefoil
and the figure-eight knots can be calculated in explicit terms.