Comparing the extent to which closure methods recover the knot
type of a closed knot from open subarcs
Emily Vecchia and Dr. Eric J. Rawdon ! University of St. Thomas
Closure methods
Protein structure
•  An open chain knot is a single curve in
three-dimensional space with two
endpoints.
Predicting knot type
Infinite rays
•  Given a subchain from a closed knot of known type, the graph
below shows the percentage of the time each closure method
correctly identifies the original knot’s type.
closure method1
•  Numerous physical systems such as
polymers, DNA, and proteins contain
open chain knots.
Open chain
•  This method extends parallel rays
from the endpoints of the given
subchain.
•  Understanding
the
connection
between structure and function in
these physical systems requires a
definition of open chain knotting.
Closed knot
•  The knot is closed by connecting these rays at infinity. In practice,
it is sufficient to connect via a line segment when the rays are
“outside” the knot.
Protein folding: To knot or not
to knot?, Eugene Shakhnovich,
January 2011.
Problem and goal
•  Classical knot theory focuses on knotted loops with no endpoints.
Open chain knotting is not yet defined mathematically.
•  The goal is to analyze and compare different methods researchers
have used for classifying knots in open chains by determining how
well each closure method predicts traditional knot theory.
Methodology
•  Each tested open chain is taken as a subchain from a closed knot
and classified using each closure method. These classifications are
then used to see how well each closure method predicts the knot
type of the closed knot from which the open chain came.
•  The process begins with the generation of 100 random closed
100-edge polygons in three-space for 23 different knot types.
•  The knotting in each open subchain (10,000 for each polygon) is
classified using the different closure methods.
•  The knot type of the closed polygon is compared to the knot
types of its subchains when they have been closed using each of
the closure methods.
•  We close in 100 directions per subchain, and these 100 closures
are used to estimate the probability distribution of knot types in
all possible directions.
Random equilateral arc
•  Dashed lines: the closure method’s guess is the most prevalent
knot type and it appears over 50% of the time.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Average of returning the knot type of the original knot
Infinite 3.1
Infinite 3.1 50%
Random arc 3.1
Random arc 3.1 50%
Infinite 9.21
Infinite 9.21 50%
Random arc 9.21
Random arc 9.21 50%
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Number of edges
closure method
•  The philosophy behind this method is that
there is a portion of the knot that is
“missing.”
•  Both methods have better performance on the 3.1 knot, a much
simpler knot than the 9.21.
•  When only considering the most prevalent knot type, the two
methods are practically the same.
•  To close the knot, a random equilateral arc is created and
concatenated with the subchain. The arc has the same number of
edges as the portion removed from the closed knot from which
the subchain came.
•  For both methods, the dashed lines lie slightly below the solid,
indicating that when a method’s closures return the correct knot
type most often, it is usually found by at least 50% of the closures.
•  The knot is closed 100 times to estimate the distribution of knot
types from all possible random closures with this number of
edges.
Distribution difference
Future work
•  In practice, one may not know the amount of edges “missing” for
a subchain. As is, the random arc method cannot be used in this
situation. We would like to test ways of estimating the edges
missing.
•  We will test a variation of the infinite method in which the rays
are extended in different directions.
Acknowledgements
A closed knot and two of its subchains
•  Solid lines: the closure method’s guess is the most prevalent knot
type returned out of the 100 times the subchain was closed.
Thanks to Dr. Eric Rawdon, Erin Brine-Doyle, Madeline Shogren,
Dr. Rob Scharein (KnotPlot), and Dr. Jason Canterella (plCurve).
1 Sułkowska,
J. I., E. J. Rawdon, K.C. Millett, J. N. Onuchic, and A. Stasiak, Proc Natl
Acad Sci USA 109(2012): E1715-23. 4.
•  The least squares difference between the distributions of closed
knot types for each subchain was computed and averaged over
number of edges and original knot type.
•  For both types, the methods are closest near 90 edges.
•  At low numbers of edges,
the distance is due to the
fact that the infinite method
will always return the
unknot, which is not the
case for the random arc
method.
•  At high numbers of edges,
the random arc is almost
always correct, so the
change in difference comes
from the infinite method.
Average distance between subchain classification
0.3
Average least squares distance
Motivation
Results, statistics, and analysis
Average of finding original knot
Type
Introduction
0.25
0.2
3.1
0.15
9.21
0.1
0.05
0
1
11
21
31
41 51 61 71
Number of edges
81
91