  The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose
vertex sets represent isotopy classes of unoriented knots, and whose edge sets
record whether pairs of knots are related by crossing changes or H(2)-moves,
respectively. We investigate quotients of these graphs under equivalence
relations defined by several knot invariants including the determinant, the
span of the Jones polynomial, and an invariant related to tricolorability. We
show, in all cases considered, that the quotient graphs are Gromov hyperbolic.
We then prove a collection of results about the graph isomorphism type of the
quotient graphs. In particular, we find that the H(2)-Gordian graph of links
modulo the relation induced by the span of the Jones polynomial is isomorphic
with the complete graph on infinitely many vertices.
