Revisiting the Second Vassiliev (In)variant for Polymer Knots

Abstract

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.

Figures (8)

• Figure 1: a. Open knotted polymers and their knot types as determined by closure. The open circles surrounding each chain represent the knot type as determined by connecting both ends to the 20 corners of a dodecahedron. The filled circles on the chain ends represent the knot type determined by connecting them directly. Left: an unconfined polymer untying between a strong $5_1$ and $3_1$ topology, with 15/20 projected closures and the end closure yielding $5_1$. Right: A confined polymer that is weakly knotted, with 8/20 projected closures yielding $5_1$ and 7/20 projected closures plus the end closure yielding $6_2$. The radius of the closure points was much larger than in the visualizations. Midset shows an open trefoil knot closed by projection to one corner of a dodecahedron. b. The typical untying sequence simulated in this manuscript, in which a $7_1$ unties into a $5_1$ then into a $3_1$ and finally into the unknot.
• Figure 2: a. Scatter plot of measure Vassiliev parameter $V$ against the true value for ideal and torus knots. The solid line represents perfect agreement. b. Steady-state Vassiliev parameters for closed $0_1$, $3_1$, and $4_1$ knots simulated as polymers. The darker data curves represent semiflexible chains ($\ell_{p}=10\sigma$) and the lighter curves represent flexible chains ($\ell_{p}=1\sigma$). The expected values are shown as horizontal lines.
• Figure 3: Mean computation time required to compute the Vassiliev parameter and the Alexander characteristic of four knots as a function of their size. Parallel dashed lines approximate the Alexander computation if sign determination and sphere projection is used. Error bars, when visible, represent the standard error on the mean, and are smaller than the symbols otherwise.
• Figure 4: Time series of the Vassiliev and absolute Alexander parameters of knots as they untie and retie. a. A knot initialized as a $7_1$ and untying through a sequence of $5_1$ to $3_1$ to the unknot. The expected values are 7-5-3-1 for the absolute Alexander polynomial and 6-3-1-0 for the Vassiliev parameter. The red curve shows the Alexander parameter as determined by minimally interfering closure, and the blue shows the average value when the ends are projected to 20 evenly spaced points on a large enclosing sphere. b. The Vassiliev and Alexander parameters during an event in which an unknotted polymer spontaneously forms a trefoil knot. In this case, the Alexander value has been rescaled so that it is easier to compare it to the Vassiliev parameter by eye.
• Figure 5: Vassiliev parameter of populations of $3_1$ and $4_1$ polymer knots untying, aligned at the time at which their minimally-closed Alexander polynomial shifts (vertical line). The population average is overlaid.