  The art of tying knots is exploited in nature and occurs in multiple
applications ranging from being an essential part of scouting programs to
engineering molecular knots. Biomolecular knots, such as knotted proteins, bear
various cellular functions and their entanglement is believed to provide them
with thermal and kinetic stability. Yet, little is known about the design
principles of naturally evolved molecular knots. Intra-chain contacts and chain
entanglement contribute to folding of knotted proteins. Circuit topology, a
theory that describes intra-chain contacts, was recently generalized to account
for chain entanglement. This generalization is unique to circuit topology and
not motivated by other theories. In this paper, we systematically analyze the
circuit topology approach to a description of linear chain entanglement. We
utilize a bottom-up approach, i.e., we express entanglement by a set of 4
fundamental structural units subjected to 3 (or 5) binary topological
operations. All knots found in proteins form a well-defined, distinct group
which naturally appears if expressed in terms of these basic structural units.
Prime knots, which are viewed by knot theory as undecomposable, are also made
of these structural units connected in some specific way. In turn, this kind of
connection shows the fundamental reason why prime knots cannot be decomposed in
the rigorous sense of knot theory. We believe that such a detailed, bottom-up
understanding of the structure of molecular knots should be beneficial for
molecular engineering.
