  We define a metric filtration of the Gordian graph by an infinite family of
1-dense subgraphs. The n-th subgraph of this family is generated by all knots
whose fundamental groups surject to a symmetric group with parameter at least
n, where all meridians are mapped to transpositions. Incidentally, we verify
the Meridional Rank Conjecture for a family of knots with unknotting number one
yet arbitrarily high bridge number.
