  Knots in open strands such as ropes, fibers, and polymers, cannot typically
be described in the language of knot theory, which characterizes only closed
curves in space. Simulations of open knotted polymer chains, often
parameterized to DNA, typically perform a closure operation and calculate the
Alexander polynomial to assign a knot topology. This is limited in scenarios
where the topology is less well-defined, for example when the chain is in the
process of untying or is strongly confined. Here, we use a discretized version
of the Second Vassiliev Invariant for open chains to analyze Langevin Dynamics
simulations of untying and strongly confined polymer chains. We demonstrate
that the Vassiliev parameter can accurately and efficiently characterize the
knotted state of polymers, providing additional information not captured by a
single-closure Alexander calculation. We discuss its relative strengths and
weaknesses compared to standard techniques, and argue that it is a useful and
powerful tool for analyzing polymer knot simulations.
