  Semiflexible polymers are widely used as a paradigm for understanding
structural phases in biomolecules including folding of proteins. Here, we
compare bead-spring and bead-stick variants of coarse-grained semiflexible
polymer models that cover the whole range from flexible to stiff by conducting
extensive replica-exchange Monte Carlo computer simulations. In the data
analysis we focus on knotted conformations whose stability is shown to depend
on the ratio $r_b/r_{\rm min}$ with $r_b$ denoting the equilibrium bond length
and $r_{\rm min}$ the distance of the strongest nonbonded interactions. For
both models, our results provide evidence that at low temperatures for
$r_b/r_{\rm min}$ outside a small range around unity one always encounters
knots as generic stable phases along with the usual frozen and bent-like
structures. By varying the bending stiffness, we observe rather strong
first-order-like structural transitions between the coexisting phases
characterized by these geometrically different motifs. Through analyses of the
energy distributions close to the transition point, we present exploratory
estimates of the free-energy barriers between the coexisting phases.
