  Knots on ring polymers tend to be tight. For knots on charged polymers
(polyelectrolytes), electrostatic repulsion among the monomers is considered to
enhance the persistence length and further localize the knots. However, the
effects of counterion condensation on knots behavior are not clear. Here we use
molecular dynamics simulations to systematically study the effect of counterion
condensation on the knot behavior under different electrostatic strength and
solvent quality conditions, with the focus on the knot sizes. We show that
generally counterion condensation \emph{delocalizes} the knots in systems with
strong electrostatic strength. At small to intermediate electrostatic strength,
the knot size and overall chain size and can be positive correlated or negative
correlated, as a result of the interplay between electrostatic correlation and
solvent quality.
