Temporary and permanent entanglements are a basic ingredient in multichain
polymer melts, and mechanically or chemically linked polymer networks
(gels,
Olympic gels, rubber - vulcanisation), respectively. Recently, there
has
been increasing interest in permanently entangled SINGLE polymer chains,
especially fuelled by their relevance in biological systems. Thus,
one
frequently encounters KNOTTED states in closed DNA, due to which
the acquisition of the genetic code during transcription is impeded.
To
overcome this, the topology is actively changed through certain
enzymes. To (further) understand how these detect a knot, a key
question is whether all topological details of the DNA-knot are localised
within a small portion of the chain (tight knot), or not.
We study the equilibrium shapes of prime and composite knots
confined to 2D. Using scaling arguments, we show that due to self-avoiding
effects, the topological details of prime knots are localised on a
small portion of the large ring polymer. Within this region, the original
knot configuration can assume a hierarchy of contracted shapes, the
leading
one given by just one small loop. This hierarchy is shown in detail
for
the flat trefoil knot, and corroborated by Monte Carlo simulations.
We also consider the force-extension curves of chains with topological
constraints.