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ACTA PHYSICA POLONICA A No. 3 Vol. 138 (2020)
Revision of Commonly Used Loop Knots Efficiencies
J. Šimona,∗, V. Dekýšband P. Palčekc
aDepartment of Applied Mathematics, Faculty of Mechanical Engineering, University of Žilina,
Univerzitná 8215/1, 010 26 Žilina, Slovakia
bDepartment of Applied Mechanics, Faculty of Mechanical Engineering, University of Žilina,
Univerzitná 8215/1, 010 26 Žilina, Slovakia
cDepartment of Materials Engineering, Faculty of Mechanical Engineering, University of Žilina,
Univerzitná 8215/1, 010 26 Žilina, Slovakia
Received: 15.11.2019 & Accepted: 24.04.2020
Doi: 10.12693/APhysPolA.138.404 ∗e-mail: jan.simon@fstroj.uniza.sk
In a number of professions, human life hangs on a knotted rope. However, until now only a poor
attention of scientists has been paid to the properties of knots. The main objective of the presented
research is to provide an in-depth revision of commonly used loop knot efficiencies employing modern
experimental technologies and correct statistical processing. In the first part of the paper, the common
mistakes in the available information sources were pointed out and the correct way of assessing the loop
knot efficiency was proposed. Subsequently, correct statistical calculus was derived to evaluate mean
knot efficiency and confidence interval. Efficiencies of eight commonly used loop knots loaded in several
geometries were precisely measured, evaluated, and analyzed. Special attention was paid to avoid
misleading conclusions based on experiments of low statistical power. Loop knot efficiency is not
a constant, but it depends at least on the static breaking strength of a rope. The process of knot breakage
was recorded by high-speed infrared thermal imaging. Analyses showed that the temperature of the most
exposed parts of a knot could reach polyamide melting point. Finally, the microfilament analysis using
electron microscopy was carried out to understand the breakage process on the microscopic level.
topics: knot efficiency, rope, tensile tests, thermal imaging, electron microscopy
1. Introduction
“Simply placing a knot in the rope before loading
it will reduce its strength and, apart from the case
of localized damage, the rope will always break at
the knot. This weakness varies from 30 to 50%, ac-
cording to the kind of knot used.” The author of
this quotation is G. Marbach [1], a French pioneer
of speleo-alpinism, and it briefly explains the rea-
sons that led to the work on the submitted study
between 2014 and 2019. A rope without a termina-
tion, respectively without an opportunity to form
a solid connection between a rope and a manipu-
lated object or a person, is in the most cases of
no use. Technically undemanding, reversible, easy,
fast, and probably the most frequently used way
how to make a rope termination is to tie a loop
knot. Unfortunately, only a few climbers, speleolo-
gists, workers at heights and mountain rescuers re-
alize how a presence of a knot fundamentally affects
rope breaking strength.
Available information sources represented espe-
cially by mountaineering textbooks, working guide-
lines, selected technical standards, specialized web-
pages, electronic articles and, to a certain extent,
even scientific papers, show that the issue of a rope
strength reduction due to presence of knot has been
dealt with for a long time (for early experimental
results see the following works [2, 3]). Later on,
works that dealt with the impact of various rope
types and used materials on knot efficiency were
written [4–10]. Among the first theoretical mod-
els that described friction and force distribution
of certain knot types were works [11, 12]. These
works were further supported by more precise mod-
els that focused also on the concentration of stress in
the cross-sectional rope area, the mutual interaction
of selected knotted structures [13–15], respectively
rope contact with different surface types [16].
Several significant changes have taken place in
rope production since the old adventurous and pio-
neering times. Producers have moved from the ap-
plication of natural construction materials to syn-
thetic materials with various surface modifications.
The reason for this was mostly biological degra-
dation of natural organic materials and a lower
strength/mass ratio [1, 17]. However, water, UV ra-
diation, and mechanical strain still affect the prop-
erties and durability of modern ropes [18–21].
The shift towards synthetic materials and impreg-
nation can be regarded as a crucial technological
progress in rope making.
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The 100 years anniversary of the Polish Physical Society — the APPA Originators
Even the construction of low stretch ropes has
been dramatically transformed, as a typical twisted
rope has been substituted by so-called “kern-
mantle rope”. Configuration with central load-
bearing part “core”, consisting of several twisted
strands of evenly distributed chirality Sand
Zsurrounded by “sheath” with both protective
and bearing functions, improves knotability, and
at the same time improves durability and safety of
the rope [1, 22, 23].
It is evident that modern ropes substantially dif-
fer from the ropes that were used by the older gener-
ation of climbers, speleologists, and rescue workers.
Due to this change, the knot efficiencies required
a general revision.
Within the previous 20 years, several works have
been published with the ambition to fill the gap
[24–34]. Many methodological guidelines and moun-
taineering textbooks have adopted the results of
these works, unfortunately, including many of the
errors. In the following text, we will try to point
out frequent mistakes that can be found across the
information sources. Namely,
•A small number of repeated experiments is
usually one of the biggest problems. Results
of our research clearly show that if we test
the strength of the same rope repeatedly (with
or without a knot), breaking strength may
vary up to 40% of the average value.
•In most cases, incorrect mathematical meth-
ods are used to calculate knot efficiency. De-
pending on the type of rope, it can yield re-
sults with margin error up to 10%.
•Imprecise or missing documentation of ex-
perimental setup, conditions and performance
of experiments is a common mistake. Inter-
pretation of such results is disputable, and
a systematic error of unknown magnitude may
skew the results.
•The photographic and image documentation
in the published works reveals that knots were
not always dressed correctly. Although it is
well known that stress distribution within un-
dressed knot is not optimal, a simple or even
multiple crossing is a common mistake.
•Authors do not distinguish between a loop
knot tied in geometry I and O (see Sect. 3.3).
This happens even though there exist works
assuming geometry I and geometry O are
not equally efficient. Moreover, we may find
cases in which the authors did not distin-
guish between standard load and cross-load
of a loop knot.
•Unclear precision of experimental setup and
absence of certifications on regular calibra-
tions may affect results by an error of un-
known significance.
•Some works do not distinguish between a fail-
ure of a knot by rope breakage and by untying,
while both failures are of a different nature
and absolutely incomparable.
Fig. 1. A graphical comparison of loop knot effi-
ciencies (data acquired from available information
sources).
•Most authors have focused on a narrow range
of knots. Only a few authors applied a uni-
form methodology in the whole range of
knots that are used in mountaineering, speleo-
alpinism, and mountain rescue. Results that
were consequently summarized in most of
the textbooks are of poor importance as these
results come from various sources. However,
the results cannot be combined due to differ-
ent methodologies of measurement, heteroge-
neous rope materials, and statistical process-
ing applied.
•Available results are extremely heterogeneous.
Their dispersion is so wide-ranged that even
the accumulation of all the above-stated er-
rors cannot explain it (see Fig. 1). One of
the most important conclusions of the pre-
sented research is that the knot efficiency is
most likely not a constant, but it at least de-
pends on the static strength of a rope. It is
clear that combining incompatible results of
different experiments measured on different
ropes must lead to an output of low informa-
tive value.
The submitted study has the ambition to avoid
the above-mentioned mistakes and to provide
the most objective information on a wide range of
loop knots in all possible variants of load, tied on
the latest generation of static ropes. Tests were per-
formed using a uniform methodology, harmonized
with existing standards when possible. Only certi-
fied and calibrated experimental setup was applied,
an emphasis was laid on correct tying and dress-
ing of knots and thorough photographic documenta-
tion. A number of experiments were selected in such
a way to use experimental material most effectively
and to get results of the highest possible statistical
power. Measurements on older static ropes were
also carried out to assess the effect of rope ageing
on knot efficiency. The points with the increased
concentration of stress and excessive friction were
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TABLE I
Parameters of tested ropes declared by the manufacturer, their mission, and stage of wear.
Manufacturer GILMONTE BEAL EDELWEISS LANEX EDELRID LANEX
Rope trademark label Profistatic
10.5 A
Contract
10.5 Bud 10.5 TENDON
Static 11 mm
Superstatic
10.5 mm
TENDON
Static 9 mm
Abbreviation G B EW L1 ER L2
EN standard Type EN 1891
Type type A rope type B rope
Material Polyamide
Year of manufacture 2015 2006 2014 2006 2008 2004
Diameter [mm] 10.5 10.5 10.5 11.0 10.5 9.1
Static breaking
strength (Tenacity) [kN] 32 25 24 34.6 30 21.3
Static breaking strength with
knot termination [kN/3min] 15 16 17 15 N/A 15
Weight per meter [g/m] 65.8 66 67 79.6 71 52.6
Impact force f= 0.3[kN] 5.9 5 5.6 5.55 N/A 4.12
Number of falls
due to EN 1891 (f= 1)11 12 12 38 N/A 11
Sheath slippage [mm] 0 0 0.2 −20 0
Elongation [%] 3.4 2.9 3.1 3.7 4 4.9
Sheath mass [%] 37 39 36.4 41.8 45 49.2
Core mass [%] 63 61 63.6 58.2 55 50.8
Shrinkage according
to EN 1891 [%] 3.8 3.3 5 3.5 2.4 3.6
Knotability 0.7 N/A 1.05 1.18 0.8 1.07
Core structure 7S+7Z 5S+6Z 7S+7Z 8S+8Z 7S+6Z 4S+5Z
Rope mission and history New, unused
Work at height
painting of vertical
constructions
Speleo-rescue Speleo-rescue Mountain
rescue
Wear stage New, unused
Both sheath
and core affected
by colour
Sheath worn, no defects
penetrating to the core
Rope length [m] 1010 100 60 200 90 50
identified using high-speed infrared thermal imag-
ing at the critical moments of knot breakage. A ma-
terial analysis using electron microscopy was carried
out to understand the principle of knot breakage on
the structural level.
Terminology denoting knots, their parts, and di-
vision into groups is derived from works [35–38].
Stand — standing part is the rope or ropes that
emerge from a knot and are loadbearing [36]. It is
the long, unknotted segment of the rope, also re-
ferred to as the dead end, standing end and the dor-
mant or stagnant part of the rope or line [38]. Wend
— working end is the short rope segment, that
emerges from a knot and is not intended to be load-
bearing [36]. It is also the terminus of the cord
employed to tie a knot [38].
Bight is a doubled-up section of rope. Knots tied
on the bight are tied using a doubled-up section
of the rope, often to produce two loops from a knot
that normally only produces a single loop. Tying on
the bight can also put a knot in the middle of a rope
without the need to access to the ends of a rope [36].
Confusingly, bight also refers to a simple wrap or
loop in a rope or cord, which can be an unfinished
knot or a portion of a completed knot [38].
Dressing is the term used for the final arrange-
ment of a knot into the correct pattern before it
is loaded. Dressing is vital to make sure that
the knot behaves, and has the breaking strength, as
expected. Dressing a knot involves not only align-
ing loops and twists but also tightening the loops
in the knot [36].
2. Material and methods
2.1. Ropes
Knots were tied on modern polyamide low stretch
kernmantle ropes (EN 1891 standard type A and B).
To ensure objectivity, both old and new ropes of
various producers were tested. However, the ropes
were made of the same material and manufactured
by similar technology. Although, the age itself
should not significantly affect the rope reliability if
it is stored correctly [19], majority of mountaineer-
ing authorities, together with most of the produc-
ers, regard the rope age as a key feature determin-
ing its performance [39–46]. Rope’s lifespan is also
affected by the intensity of rope usage, especially
top-rope climbing and rappelling [47]. Other factors
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TABLE II
Parameters of tested accessory cord declared by the
manufacturer, its mission, and stage of wear.
Manufacturer GILMONTE
Accessory cord trademark label Rep 8 mm
Abbreviation G8
EN standard EN 564
Material Polyamide
Year of manufacture 2015–2018
Diameter [mm] 8
Static breaking strength (Tenacity) [kN] 18
Weight per meter [g/m] 40.2
Core structure 3S+3Z
Mission and history New, unused
Wear stage New, unused
Rope length used for experimental
examination [m]
562
are UV radiation, abrasive particles, acids and ag-
gressive chemicals, temperatures above 50◦C, and
weather conditions [21, 48, 49].
The age of ropes used in the presented paper
ranged from 1 to 12 years. The tested ropes were
previously used for speleo-rescue, mountain rescue,
and work at height.
Detailed characteristics of the tested ropes ac-
cording to the EN 1891 standard and accessory
cords according to the EN 564 standard are shown
in Tables I and II.
2.2. Tensile tests
Breaking strength of ropes and knot efficiencies
were tested using a horizontal testing device de-
signed for static tests of climbing ropes. It was pro-
duced in 2006 by Engineering Test Institute, Public
Enterprise (SZU), Brno. The device is mounted in
ropemaking company Gilmonte, in a room with per-
manent laboratory conditions. Force measurements
were performed by tensometric sensors connected
to the control unit and a computer. Two oppo-
site cylindrical self-locking jaws with the diameter
of 120 mm were used to fix the rope. The rope was
wrapped three times and pulled through the cen-
tral gap, based on the principle of tensionless hitch.
Hence, the rope was anchored purely by friction,
gradually stretched in accordance with the Euler–
Eytelwein equation [16] and was not exposed to
strain in shear or twist. If required, one cylindri-
cal jaw was used against which a smoothly pol-
ished steel ring simulating a climbing carabiner was
placed. Jaw movement was provided by a hydraulic
mechanism within a range of 0–3 m. The speed of
the jaw movement was set to 180 mm/min accord-
ing to the EN 364 standard. Calibration of the ma-
chine was performed once in two years by the inde-
pendent institution Slovak legal metrology. A rel-
ative measurement error is guaranteed to be lower
than 0.17% within the interval of acting forces.
2.3. High-speed infrared thermal imaging
and rope emissivity
Recordings of temperature distribution within
the knot body and surroundings were performed by
the high frame rate infrared camera FLIR SC7500
with InSb detector 320 ×256 pixels at the native
frame rate of 383 Hz. Knot breakage is a rela-
tively fast process that takes place in the course
of 10−5–10−4s. In order to record crucial stages of
knot breakage, the frame rate was set up to 1253 Hz.
Thus, individual images were taken in 0.8 ms in-
tervals. The camera worked in the superfram-
ing mode with three integrated times (from 19 up
to 433 µs), with the objective L0116 or L0106 with
the focal length 25 or 50 mm, IFOV= 0.6mrad.
The whole infrared system was mounted on a tri-
pod at a distance of 1.5–1.7 m from the breakage
zone. The field of view was thermally separated
from the surrounding environment by polystyrene
panels.
A crucial factor in the process of recorded signal
transformation to thermal imaging is the emissiv-
ity. Measurement of this quantity is nontrivial and
requires specialized experimental devices. There-
fore, we established a collaboration with the New
Technologies-Research Centre at the University of
West Bohemia. Here the emissivity was directly
measured for the whole Gilmonte Profistatic rope,
as well as for the isolated core at polar angle 10◦
and various temperatures. The method of measure-
ment of the effective directional emissivity at high
temperature (EDEHT) is described in the following
works [50, 51]. Results of emissivity measurements
are available in Tables III and IV.
2.4. Electron microscopy
The nature of individual microfilaments damage
before and after knot breakage was investigated by
the scanning electron microscope TESCAN VEGA
II LMU. Since microfilaments are electrical insula-
tors, detector LVSTD was used to scan the surface
of broken fibers without metallization.
TABLE III
Emissivity εof core measured on rope GILMONTE
Profistatic 10.5.
T[◦C] 50 100 150 180 190 200
ε0.795 0.804 0.779 0.853 0.869 0.865
Std. dev. 0.103 0.062 0.046 0.044 0.043 0.041
TABLE IV
Emissivity εof whole rope (core + sheath) measured
on rope GILMONTE Profistatic 10.5.
T[◦C] 50 100 150 180
ε0.801 0.834 0.833 0.844
Std. dev. 0.104 0.066 0.050 0.087
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The 100 years anniversary of the Polish Physical Society — the APPA Originators
3. Results and discussion
3.1. Rope without a knot
The measurement of the straight rope static
breaking strength is the first step in the calcula-
tion of knot efficiency. Symbol yhas been reserved
to label this quantity in the following text.
Statistical analysis confirmed that the static
breaking strength of a rope can be regarded as nor-
mally distributed with mean value ¯yand variance
σ2
0= (y−¯y)2. Therefore, we can write the prob-
ability density function of static breaking strength
by the following formula:
ρ0(y) = 1
√2πσ0
exp −(y−¯y)2
2σ2
0!.(1)
Experimental results proved that ¯y/(√2σ0)1
and so we can state that R∞
0ρ0dy∼
=R∞
−∞ ρ0dy= 1.
Therefore,
•Despite the fact that the negative strength
does not have any physical sense, it can
be considered as the normally distributed
quantity.
•Regardless of the age but depending on
the degree of wear, a rope is a heteroge-
neous structure with variable strength from
one place to another. Differences in static
breaking strength of adjacent parts of the
same rope can reach up to 40% of an aver-
age value. Due to this fact, any conclusions
on rope strength, with or without knots, can
be based only on a set of representative mea-
surements yielding results of high statistical
significance.
•According to rope producers, if properly
stored, rope ageing is not harmful to ropes for
a certain period [42–46]. Results of our tests
confirm these statements; properly stored old
rope performed as a new one even after
10 years.
•The static breaking strength of a rope de-
creases by rope usage. Within the performed
experiments we recorded a decrease by 7–24%
in static breaking strength when compared to
a new rope.
•Statistical analysis neither confirmed nor dis-
puted the weakening of the central and termi-
nal parts of a rope.
3.2. Rope terminated by a loop knot:
statistical properties of the knot efficiency
A rope in which a knot is tied always shows lower
static breaking strength compared to the rope with-
out a knot. The effect has been known for a long
time and the term knot efficiency or residual break-
ing strength (RBS) was adopted for its quantifi-
cation. Knot efficiency is defined as a propor-
tion of static breaking strength of a rope in which
the knot is tied (marked as x) and static break-
ing strength of the same rope without the knot
(marked as y). Knot efficiency η=x
yis usually ex-
pressed in a percentage.
Breaking strength of rope with or without a knot
can be measured directly, but knot efficiency can-
not. That is why we must evaluate knot efficiency
by a calculation.
Statistic tests confirmed that static breaking
strength of the rope with a knot termination is nor-
mally distributed around the mean value ¯xwith
a variance σ2= (x−¯x)2. Therefore, we can write
the probability density function of the static break-
ing strength of rope terminated by a knot by the fol-
lowing formula:
ρ(x) = 1
√2πσ exp −(x−¯x)2
2σ2!.(2)
Experimental results proved that ¯x/(√2σ)1
and so we can state that R∞
0ρdx∼
=R∞
−∞ ρdx= 1.
Therefore, despite the fact that the negative
strength does not have any physical sense, it can
be considered as the normally distributed quantity.
It can be shown that the knot efficiency falls
within the interval hη, η + dηiwith probability
Phη,η+ dηi=R∞
0R(η+ dη)y
ηy ρρ0dxdy, where ρand
ρ0are defined by (1) and (2). After evaluation
of inner integral we get the probability density
function of knot efficiency
ρη(η) =
∞
Z
0
yρ (ηy)ρ0(y) dy. (3)
Distribution ηρis not symmetric (see Fig. 2) and
it is clear that the mean knot efficiency cannot be
calculated by the relation ¯η=¯x
¯yas is widely used by
authors, but by integral ¯η=R∞
0ηρη(η) dη. Using
Monte Carlo method, we estimated that incorrect
technique of the calculation can lead to an error up
to 5–10% when compared to formula (3).
Fig. 2. An example of the probability density
function of the standardly loaded figure eight loop
(geometry I) given by (3). Black line represents
mean value; 68.27% of all measurements fall within
the blue shaded interval.
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The 100 years anniversary of the Polish Physical Society — the APPA Originators
Exactly 68.27% of all results fall within
the confidence interval hηL, ηHi, where
RηL
0ρη(ξ)dξ= 15.87% and R∞
ηHρη(ξ)dξ= 15.87%.
Interval hηL, ηHiis a non-symmetric analogy to the
1−σconfidence interval of normal distribution.
3.3. Loop knots in general
Loop knot is a knotted structure that features
one or more loops. Loops can be fixed, slippery
or adjustable. Some slip loops can collapse down to
a simpler knot or, if not attached to another object,
become untied entirely when loaded. In general,
a loop maintains its structure even when removed
from its point of attachment [38]. Loop knots can
be loaded under standard load, cross load and ring
load [52–54].
When a loop knot is under standard load, a pair
of opposing and equal forces act on the stand and
the loop. Vectors of both forces lie on a straight line
which is usually equal to the longitudinal knot sym-
metry axis. Standard load is an optimal way to load
a loop knot as it resists both untying and breakage
better than under any other conditions. There are
two possible ways how to tie the vast majority of
loop knots, thus let us call them geometry I and
geometry O (Fig. 3). They can be characterized as
follows:
•In geometry I, the stand and the wend twist
around the loop in such a way that the stand
is on the inside and the wend on the outside of
the undressed collar. Stand is proximal and
wend distal to the loop on the dressed loop
knot;
•In geometry O, the stand and the wend twist
around the loop in such a way that the stand
is on the outside and the wend on the inside of
the undressed collar. Stand is distal and wend
proximal to the loop on the dressed loop knot;
•When loop knot is under cross-load, a pair of
opposing and equally powerful forces acts on
the wend and the stand. There are additional
strain components in the rope that are not
present in the standardly loaded knot. That
is why the efficiency of cross-loaded loop knots
is significantly lower. Loop is usually loaded
by negligible small force when compared to
forces acting on stand and wend. If knot has
the longitudinal axis of symmetry, then acting
forces lie on the straight line which is approx-
imately perpendicular to it (Fig. 3);
•When loop knot is under ring-load, two or
more forces act on the loop so that their vec-
tor sum is zero. Approximately half of the
acting force is transmitted to the knot. Part
of the ring-load can be dissipated by friction
between the loop and anchoring object.
It is also possible to load a loop knot by the com-
bination of the above-mentioned loads. However,
we did not consider these in our research.
Fig. 3. Standardly loaded figure eight loop
(geometry I and O).
In the presented study, we have experimentally
investigated the following loop knots: figure eight
loop, double figure eight loop, figure nine loop, over-
hand knot, bowline, left-hand bowline, bowline on
a bight and alpine butterfly knot, which are among
the most commonly used knots in their category.
3.3.1. Figure eight loop
Figure eight loop (Flemish loop, ABOK #1047)
is one of the most widely used single-loop knots.
Therefore, we set it as a referential benchmark in
the following text.
The vast majority of methodological materials
recommend it as a first choice how to attach a rope
to a harness (so-called tie-in) because of its rela-
tive simplicity, reliability, symmetry, and easy vi-
sual partner check. Real-life situations have proved
that properly tied figure eight loop is more reliable
than more complicated knots tied in the wrong way.
It is used for multiple purposes, but mainly to tie-
in, to form a rope termination for single loop an-
choring in mountaineering, mountain rescue, speleo-
alpinism and work at height. It is recommended to
tie this knot using ropes with a diameter of at least
9 mm as it can be hard to untie after heavy loading
or fall arrest [1, 40, 41, 55–60].
It can be tied on a bight (using a carabiner or
a post), as well as at the end of a rope around
the closed anchoring point (so-called rewoven or
rethreaded figure eight, tied by rethreading figure
eight stopper ABOK #524 in reverse).
It is possible to load it standardly in both geome-
tries I and O (Fig. 3), under cross-load (Fig. 4) or
ring-load. Standardly loaded figure eight loop in
geometry I is also recognized by EN 1891 and EN
892 as a rope termination for tensile tests. It does
not tend to untie under cyclic loading. Figure eight
loop was tested under standard loading (in both ge-
ometries I and O) and cross-load in the presented
study (see Fig. 5). One can observed that
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The 100 years anniversary of the Polish Physical Society — the APPA Originators
Fig. 4. Cross-loaded figure eight loop.
Fig. 5. Figure eight loop efficiency.
•The efficiency of standardly loaded figure
eight loop in O and I geometry is statisti-
cally indistinguishable at the significance level
5%. Therefore, we can consider the effi-
ciency in both geometries to be the same.
This conclusion is supported by a total of 81
measurements using five different ropes, and
it is in direct contradiction to the following
works [7, 8, 36, 56, 61, 62].
•Figure eight loop is one of the most widely
used knots. Therefore, we paid special at-
tention to its experimental examination. We
measured its efficiency in standard loading
on 11 different ropes of comparable diameter.
We found out that the results differ up to
16%, although experimental conditions were
identical and the experimental setup error
was negligible. In order to explain this phe-
nomenon, we correlated the efficiency with
the different parameters of the rope on which
the knot was tied. An interesting result was
found when we related the efficiency to the
static breaking strength of the rope (Fig. 6).
In all likelihood, the efficiency of the stan-
dardly loaded figure eight loop is a decreasing
function of the static breaking strength of the
rope. Under first-order approximation this
function can be linearized by
η(y)∼
=−0.77 (y−32.79) + 66.45%
(correlation coefficient R2= 0.67).
Fig. 6. Standardly loaded figure eight loop effi-
ciency as a function of the static breaking strength
of the rope.
Thus, the efficiency of the standardly loaded
figure eight loop is very unlikely to be a con-
stant, contrary to what all available publications
implicitly present. This statement may also be
generalized to other loop knots, but it would re-
quire further experimental evidence. If the for-
mulated hypothesis is correct, then it makes no
sense to compare the efficiency of the knots tied on
different ropes.
This result can be indirectly supported by several
works [63, 64], which attempted to correlate various
knot efficiencies with a rope diameter. Evans [65]
in his article concluded that there is a trend for
knots in larger diameter materials to retain less of
the original unknotted strength. To make the link-
age clear, let us briefly acknowledge that the static
breaking strength of rope yand rope diameter r
are under second-order approximation coupled by
increasing relation y≈r2.
The efficiency of a cross-loaded figure eight loop
was determined on two ropes and one accessory
cord. In comparison to the standard loading, the
measured efficiency was lower by 19.26 ±0.44%. As
a rule of thumb, it should be kept in mind that the
cross-load reduces the efficiency of the figure eight
loop by ≈20%.
3.3.2. Double figure eight loop
Double figure eight loop (bunny ears, ABOK
#1085) is a knot of the figure eight loop group.
However, it has two loops available for anchoring
(see Fig. 7). As a result, it better resists ring-load,
or loop cut at high forces and small curvature radii
of the anchoring point. Another advantage is that
it is easier to untie after heavy load and to adjust
the size of the individual loops, used for example in
building a multi-point equalized anchor [55, 56, 60].
It is tied mainly on a bight using a carabiner or
a post. It is theoretically possible, but much more
complicated, to tie the knot around a closed an-
chor point. It is possible to load it standardly in
both I and O geometries, as well as under cross-
load and ring-load.
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Fig. 7. Standardly loaded double figure eight loop
(geometry I and O).
Fig. 8. Double figure eight loop efficiency.
Double figure eight loop was tested in standard
loading in both geometries I and O (see Fig. 8). One
can observed that
•The efficiency of the standardly loaded double
figure eight loop in geometry O and I is statis-
tically indistinguishable at a significance level
5%. Therefore, we can consider the efficiency
in both geometries to be the same. This con-
clusion is supported by a total of 22 measure-
ments that were performed on the same rope.
•The difference in the efficiency of the stan-
dardly loaded double figure eight loop com-
pared to the referential figure eight loop is
statistically indistinguishable at a significance
level 5%. This conclusion is supported by a
total of 44 measurements performed on the
same rope in both geometries O and I. Based
on these results, we can conclude that double
figure eight loop does not perform better than
a much simpler figure eight loop.
3.3.3. Figure nine loop
Figure nine loop (rethreaded version of ABOK
#521) is one of the newest single-loop knots belong-
ing to the figure eight knot group. However, it has
a more complex structure compared to the figure
eight knot (see Fig. 9), thus the strain is better dis-
tributed within the whole knot. After a heavy load
or fall arrest, the figure nine loop does not require
such a great effort to untie as compared to figure
Fig. 9. Standardly loaded figure nine loop
(geometry I and O).
Fig. 10. Figure nine loop efficiency.
eight loop [57]. Given the higher efficiency, it works
better with ropes of smaller diameter than the fig-
ure eight loop [55, 66]. However, figure nine knot is
slightly more difficult to tie so there is a risk of mak-
ing a mistake when tying or dressing it. Moreover,
it is necessary to count with higher rope consump-
tion [1]. Compared to figure eight loop, it fills by
approximately 10% larger volume. Thus, attention
should be paid to prevent the knot from damage by
excessive rubbing against a rock [56]. This type of
rope termination can be tied on a bight (using a
carabiner or a post) as well as by rethreading figure
nine stopper (ABOK #521) around a closed anchor
point. The knot does not tend to untie under cyclic
loading.
It is possible to load it standardly in both I and
O geometries, under cross-load, as well as ring-load.
Figure nine loop was tested under standard load
in both I and O geometry (see Fig. 10). One can
observed that
•The efficiency of a standardly loaded figure
nine loop is not the same in geometries O and
I (it is statistically distinguishable) at a sig-
nificance level 5%. The efficiency in geometry
O is about 3.75% higher compared to geome-
try I. This conclusion is supported by a total
of 19 measurements that were performed on
the same rope. In the future, it is necessary
to verify this conclusion by an independent
experiment using multiple ropes.
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Fig. 11. Standardly loaded overhand loop (geome-
try I and O).
Fig. 12. Cross-loaded overhand loop.
•Standardly loaded figure nine loop proved to
have the highest efficiency of all loop knots
presented in this study in all of the tested
geometries. When compared to a referential
figure eight loop, the difference in the effi-
ciency is ∆ηO= +10.03% and ∆ηI= +5.83%.
Based on the results, we recommend using
the figure nine loop in O geometry for anchor-
ing in the situations where high static loads
are expected (e.g. high angle rescue tech-
niques).
3.3.4. Overhand loop
Overhand loop (loop knot, overhand knot on a
bight, ABOK #1009) is one of the simplest single-
loop knots with low rope consumption and a high
degree of symmetry. Due to these benefits, it is pop-
ular and widely used. On the other hand, it is chal-
lenging to untie the knot after heavy load, especially
when soft ropes with a diameter of 9 mm or less are
used. It is usually tied on the bight in the middle
of a rope to tie-in, as a fall arresting knot on glacier
travel, as a leg loop on an accessory cord, or as a
simple accessory knot in climbing where no dynamic
stress is expected [1, 40, 41, 56, 58]. This rope ter-
mination can be tied on a bight (using a carabiner or
a post) as well as rethreaded overhand knot stopper
(ABOK #46) around a closed anchor point. This
knot is also preferred when rappelling to tie together
two ropes of the same diameter [40, 41]. In terms of
the force, we can speak de facto about cross-loaded
overhand loop knot (in this case the knot is called
offset overhand bend, ABOK #1410).
Fig. 13. Overhand loop efficiency.
Fig. 14. Standardly loaded overhand loop effi-
ciency as a function of the static breaking strength
of the rope.
It is possible to load it standardly in both I and
O geometries, under cross-load and ring-load (see
Fig. 11 and 12). Overhand loop was tested under
standard load (in both I and O geometries) and
cross-load (see Fig. 13). One can observed that
•The efficiency of standardly loaded overhand
loop in O and I geometries is not the same
(it is statistically distinguishable) at a signifi-
cance level 5%. The efficiency in geometry O
is slightly higher by 4.45% compared to geom-
etry I. This conclusion is supported by a total
of 24 measurements performed on the same
rope. In the future, it is necessary to ver-
ify this conclusion by an independent experi-
ment using various ropes. Since then we rec-
ommend to prefer overhand loop in geometry
O, at least in life-critical applications.
•Overhand loop is one of the most popular
knots; therefore, we paid special attention
to experimental examination of this knot.
We measured its efficiency under standard
load using 5 different ropes of a comparable
diameter. The results show that the efficiency
of the standardly loaded overhand loop to a
first approximation is a linearly decreasing
function of the static breaking strength of
the rope (see Fig. 14):
η(y)∼
=−0.62 (y−32.79) + 60.67%.
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Thus, we have another good reason to believe that
in general the efficiency of the standardly loaded
loop knots is not constant.
Compared to a referential figure eight loop,
the standardly loaded overhand loop proved to have
lower efficiency in geometry I, ∆ηI=−6.39±1.75%
at a significance level of 5%. In geometry O, the ef-
ficiency of both knots is at the resolution threshold
and its difference is ∆ηO=−3.29 ±1.10%. How-
ever, the lower efficiency was proven only on unused
ropes. For used ropes, the variance of measured
values is so extensive that the efficiency of knots
cannot be statistically distinguished. These conclu-
sions are supported by a total of 108 measurements
that were performed on five different ropes in both
O and I geometry. The efficiency of the cross-loaded
figure eight loop and overhand loop is statistically
indistinguishable at a significance level of 5%. This
conclusion is supported by a total of 48 measure-
ments that were performed on three different ropes.
Based on the findings, the overhand loop is more
suitable in situations when static cross-load is ex-
pected (tying-in at glacier travel) thanks to easier
tying and lower rope consumption.
3.3.5. Bowline
Bowline (single bowline, bowling, ABOK #1010)
is a single-loop knot and can be used for single-
point anchoring (see Fig. 15). In the past, it has
been used for tie-in to a harness. It is characterized
by low rope consumption, simplicity, and speed of
tying. Furthermore, it is possible to tie the knot
with one hand.
Unlike the knots of the figure eight group,
the bowline can be easily untied even after being
exposed to an intense load. The wend does not
rethread the basic stopper structure and I or O ge-
ometry is not defined. The main disadvantage of a
bowline is that it is not inherently secure. It means
that it can untie at relatively low forces when ring-
loaded, therefore stopper knot is mandatory [58].
Furthermore, it tends to spontaneously loosen un-
der cyclic loading and can be tricky to inspect as it
is not symmetric.
For the reasons mentioned above, it is not rec-
ommended to use this knot in life-critical applica-
tions anymore [47, 55, 56, 60, 67]. With a heavy
rope strain, a bowline knot can capsize (twist to
a position in which it is easy to untie). However,
if the knot is tied correctly, there is little or no
danger of capsizing before the breaking point of
Fig. 15. Standardly loaded bowline.
Fig. 16. Bowline, left-hand bowline, and bowline
on a bight efficiency, standard load.
Fig. 17. Force applied over time by ring-load to
bowline and left-hand bowline.
the rope itself is reached [35]. Bowline can be rea-
sonably tied only at the end of a rope and therefore
was tested under standard load and ring-load (see
Figs. 16 and 17).
3.3.6. Left-hand bowline
Left-hand bowline (cowboy bowline, ABOK
#1034½) is a single-loop knot and it differs from
the bowline by having wend on the outside of
the loop (see Fig. 18). It has significantly better
properties under ring-load than the bowline knot.
The knot has a tendency to spontaneously loosen
under cyclic loading [67]. Various attributes are
assigned to this knot even in respected literature.
However, validity of these attributes is at least dis-
putable. Similarly to bowline, it is not inherently
secure and therefore not recommended to use in life-
critical applications.
Left-hand bowline was tested under standard
load and ring-load (see Figs. 16 and 17).
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3.3.7. Bowline on a bight
Bowline on a bight (ABOK #1080) is a double-
loop knot of the bowline group (see Fig. 19). The
advantage is its higher reliability and the possibil-
ity to adjust the size of individual loops for multi-
point anchoring. Compared to the double figure
eight loop, it is more compact, the rope consump-
tion is lower, and it is easy to adjust the loop size [1].
It is tied mainly on a bight using a carabiner or a
post. It is possible, but much more complicated, to
tie it around a closed anchor point.
It is possible to load it standardly in both geome-
tries I and O, both under cross-load and ring-load.
Bowline on a bight was tested under standard load
in geometry I (see Fig. 16). One can observed that
•The efficiency of standardly loaded bowline,
left-hand bowline and bowline on a bight
in geometry I is statistically indistinguish-
able at a significance level 5%. Therefore,
the efficiency of all three knots can be con-
sidered the same. This conclusion is sup-
ported by a total of 51 measurements and it
is in direct contradiction with proposition in
the book [40], according to which left-hand
bowline is “much weaker”.
•All standardly loaded knots of the bowline
group have lower efficiency when compared
to referential figure eight loop at a signifi-
cance level 5%. The difference in the efficiency
is ∆η=−4.64 ±1.53%. These conclusions
are supported by a total of 101 measurements
that were done on three different ropes.
•Bowline and left-hand bowline were also
tested under ring-load. None of the knots
broke, but both of them untied. Therefore,
it is irrelevant to talk about their efficiency
when ring-loaded. The process of untying has
two phases. In the first phase, the grip of
the nipping turn is sufficient to resist the ring-
load. When the strain exceeds the threshold
(let us call it F0), the wend starts to move
and the knot begins to slip periodically. This
force has an average size F0B = 1.95±0.18 kN
for bowline and F0LHB = 10.77 ±1.82 kN
for left-hand bowline. In a short period,
due to the friction and heat, the rope sheath
gets smoother and the friction force decreases
to the level F1. The average value of this
force is F1B = 1.63 ±2.34 kN for bowline
and F1LHB = 11.33 ±0.97 kN for left-hand
bowline. As the results show, the difference
in the disintegration force between a bow-
line and left-hand bowline is statistically un-
questionable. In conclusion, the left-hand
bowline withstands the static ring-load bet-
ter than the bowline (see Fig. 17). However,
in both cases, the knots became unstable un-
der cyclic loading and untied at relatively low
forces when ring-loaded. Therefore, we do not
recommend to use these knots in life-critical
applications.
Fig. 18. Standardly loaded left-hand bowline.
Fig. 19. Standardly loaded bowline on a bight
(geometry I).
Fig. 20. Cross-loaded alpine butterfly knot.
3.3.8. Alpine butterfly knot
Alpine butterfly knot (Lineman’s knot, ABOK
#1053) is a single-loop knot. It is usually tied in
the central part of the rope and it is unique as
the wend opposes the stand (see Fig. 20). This
makes it significantly different from most of the loop
knots that are much more weakened by cross-load.
When tying, it is necessary to pay attention in or-
der to avoid tying its clone (so-called false butterfly
knot), which slips at a lower force and is suitable
as a shock-absorbing knot [1]. Due to its stability
and optimal position of wend and stand under cross-
load, alpine butterfly may be used as a tie-in knot in
a middle rope section for a glacier travel [59]. It is
also used for anchoring hand lines and fixed ropes
in traverses or disengaging shorter rope segments
when damaged. Even after the knot is loaded, it
can be untied relatively easily [60, 68] and it does
not tend to disintegrate under cyclic loading.
Alpine butterfly knot was tested only under
cross-load (see Fig. 21). Alpine butterfly knot
has the highest efficiency when compared to other
cross-loaded knots at a significance level of 5%.
This conclusion is supported by a total of 32
measurements performed on one rope and one
accessory cord.
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Fig. 21. Alpine butterfly knot efficiency.
Fig. 22. Loop knots ordered from highest to low-
est efficiency (based upon Table V). Mean value,
confidence interval 68.27%, and extreme measured
values are shown. Blue color is used for standardly
loaded loop knots in O geometry, red for I geometry
and green for cross-loaded loop knots.
Fall arresting knots on glacier travel should be
standardly placed between members of the roped
party. Instead of figure eight loop and overhand
loop [40, 60, 69] we recommend to use alpine but-
terfly. This knot performs much better under cross-
load and should be preferred for this purpose. The
difference in the efficiency compared to the referen-
tial cross-loaded figure eight loop is ∆η= +10.52%.
This conclusion is supported by a total of 18 mea-
surements on the same rope.
3.3.9. Knots comparison
Knot efficiency was measured on the ropes from
various manufacturers, of different diameters and
different stages of wear. As we already noticed, such
results are not comparable. However, all the an-
alyzed knots were tested on Gilmonte Profistatic
rope, which allows us to compare and rank the knot
efficiencies (see Table V and Fig. 22). We assume
that the results can be applied to any static kern-
mantle rope with a diameter close to 10.5 mm and
static breaking strength at the level of 30–35 kN.
TABLE V
Loop knots ordered from highest to lowest efficiency,
measured on static rope Gilmonte Profistatic 10.5.
Results were calculated by (3) and coupled into
groups. Members of each group are statistically
indistinguishable.
Loop knots
Mean
efficiency
[%]
Confidence
interval
68.27%
Difference
from figure
eight loop
Standardly loaded loop knots
Figure nine loop, O 74.00 71.19 10.03
76.81
Figure nine loop, I 70.25 67.49 5.83
73.01
Double figure
eight loop, O 65.13 61.32
68.94
Figure eight loop, I 64.42 61.87
66.98
Figure eight loop, O 63.97 62.20
65.73
Double figure
eight loop, I 62.93 59.89
65.97
Overhand knot, O 61.46 57.73 −3.29
65.21
Bowline
on the bight, I 60.95 57.12
−4.64
64.77
Left-Hand Bowline 60.00 58.57
61.43
Bowline 58.51 54.48
62.53
Overhand knot, I 56.92 52.96 −6.39
60.88
Crossloaded loop knots
Alpine butterfly, C 54.60 49.21 10.52
60.00
Overhand, C 46.12 42.34
49.91
Figure eight, C 44.08 42.11
46.05
3.3.10. Electron microscopy and thermal imaging
Polyamide microfilaments before and after rope
breakage were zoomed using electron microscopy
(see Figs. 23 and 24). Intact microfilaments are
of cylindrical shape with a constant diameter of
28.85 ±0.62 µm. On the other side, broken fibers
have recognizable thickened terminal parts with a
mean diameter of 64 ±2µm. The broken fibers
gradually decrease in diameter with increasing dis-
tance from the end. After approximately 275 µm
diameter decreases by a factor of 2. In the thick-
ened area, the surface of the fibers is covered by
flake-like structures (Fig. 24).
Tensile test from the structural point of view
can be divided into several temperature inter-
vals [70, 71]:
1. As the strain rises, rope temperature increases
from the laboratory conditions to a tempera-
ture of 50◦C. Young modulus of elasticity can
be considered constant within this interval.
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Fig. 23. Electron microscopy digital imaging; ter-
minal parts of polyamide microfilaments cut by
a scalpel.
Fig. 24. Electron microscopy digital imaging; ter-
minal parts of polyamide microfilaments torn in ten-
sile tests.
2. Additional increase of strain leads to rising
temperature within interval h50 ◦C, 100 ◦C).
Here polyamide crosses the glass-transition
temperature Tg. Within this interval Young
modulus of elasticity decreases and polyamide
passes from glass state to rubbery state.
3. Further increase of strain leads to rising
temperature within interval h100 ◦C, 205 ◦C).
The Young modulus of elasticity is even lower
than in the previous interval.
4. Temperature interval over 205 ◦C covers
polyamide melting point temperature Tm.
Within this interval, Young modulus of elas-
ticity reaches zero and rope breaks.
High-speed thermal imaging shows (see
Figs. 25–31) that the highest temperature of
standardly loaded loop knot just before the knot
breaks can be found in the entry part of the stand.
The temperature here varies within the interval
h60 ◦C, 80◦Ci, so glass-transition temperature Tgis
exceeded before breakage.
At the moment when the rope breaks, or after
approximately one millisecond after it, the surface
temperature of the most exposed parts reach values
close to Tm. Occasionally, even higher temperature
than Tmwas locally detected. At this moment we
do not have experimental proof whether the mo-
ment when the rope breaks precedes the rapid tem-
perature jump to the interval 4 or vice versa. With-
out even faster thermal imaging setup we are not
Fig. 25. Thermal imaging of standardly loaded fig-
ure eight loop breakage (geometry O).
Fig. 26. As in Fig. 25, but for geometry I.
Fig. 27. Thermal imaging of cross-loaded figure
eight loop breakage.
able to determine whether the temperature increase
causes the rope break, or the event of rope break
causes the increase in temperature. Most likely,
the truth is somewhere in the middle. At the mo-
ment, we neither have a provable explanation for
the process of the temperature jump.
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Fig. 28. Thermal imaging of standardly loaded
double figure eight loop breakage (geometry O).
Fig. 29. As in Fig. 28, but for geometry I.
Fig. 30. Thermal imaging of standardly loaded fig-
ure nine loop breakage (geometry O).
But it is reasonable to say that a significant
amount of energy is accumulated into rope just be-
fore rope break. Rope absorbs the energy by macro-
scopic stretching deformation and by change of its
microscopic structure orientation to the direction
Fig. 31. Thermal imaging of standardly loaded
overhand loop breakage (geometry I).
of applied force. As the rope breaks, relaxation
and structural reorientation takes place in an ex-
tremely short time. At the same time, rope frag-
ments move apart under high friction force gener-
ated by the remaining knot body. Both processes
can generate sufficient heat attended by the emis-
sion of microwave radiation.
The thickening of broken fiber’s ends can be ex-
pressed by the following hypothesis. At the moment
when a rope breaks, or shortly after it, the tem-
perature in the break zone jumps to the level 3
or even 4 using the mechanism described above.
Backward movement of disintegrated fragments is
highly non-inertial. Molten microfilaments are com-
pressed by a deceleration force. At the same time,
temperature quickly decreases and transition from
oriented structure to original unoriented structure
takes place. Fibers’ ends final shape is formed as
the surface becomes stable by further cooling down
of the material.
3.4. Future research
For sure, there are still many topics remaining to
be researched. Let us list just a few of them:
1. Presented study deals only with the loop
knots. There are many commonly used bends,
hitches, or friction hitches that are still uncov-
ered by serious research;
2. The presented study is focused solely on low-
stretch ropes designated mainly for work and
rescue. Knot efficiencies on dynamic climbing
ropes still has not been researched seriously
and to a sufficient extent;
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The 100 years anniversary of the Polish Physical Society — the APPA Originators
3. Measurements in the presented study were
performed at low-speed stretching machine
(the speed of jaw movement was set to
180 mm/min). Rope stretching in real-life ap-
plications is almost always faster and more
dynamic. The impact of stretching dynamics
on knot efficiency remains unclear.
4. The influence of static breaking strength of
rope on knot efficiency was proved on figure
eight loop and overhand loop. Generalization
of this conclusion to all loop knots, or even
hitches and bends, requires much more rigor-
ous experimental work;
5. Besides the static breaking strength of a rope,
there are many other parameters that may
affect the knot efficiency. Hot candidates to
be examined are rope diameter, rope wetness,
knotability, and smoothness of sheath;
6. It seems that temperature can play an im-
portant role in the process of knot breakage.
It changes so fast, that further research would
require experimental setup with order of mag-
nitude higher frame rate, i.e., 10−5–10−4s;
7. Improper knot dressing has been recognized
by many authors to have a negative impact
on the knot efficiency. However, this has never
been measured and analyzed seriously.
4. General conclusions
The efficiencies of 8 commonly used loop knots
were precisely measured, evaluated and analyzed on
various low-stretch ropes to an unprecedently large
extent. We laid emphasis on high statistical power
of performed experiments and precisely controlled
experimental methodology, which is extremely im-
portant when studying such a heterogeneous struc-
ture like a knotted rope. Statistical apparatus for
mean knot efficiency and confidence interval evalua-
tion was derived and applied. Properties of studied
knots were revised and their mutual comparison was
carried out (see Sects. 3.3.1–3.3.9). With a proba-
bility bordering on certainty, we may conclude that
the efficiency of loop knots is not constant as it
is implicitly presented across the vast majority of
published works, but it is a decreasing function of
static breaking strength of the rope. Furthermore,
electron microscopy and high-speed thermal imag-
ing revealed that knotted rope is subjected to tem-
peratures of extraordinary extent as the knot breaks
(Sect. 3.3.10). Approximately one millisecond after
knot breaks, the surface temperature of the most
exposed parts can reach a temperature close to the
polyamide melting point.
Acknowledgments
This work would not have been possible without
the material and technical support of the ropemaker
company Gilmonte, Slovak mountain rescue service
(HZS) and the University of Žilina.
The work was financially supported by Scientific
Grant Agency of the Slovak Republic under grant
VEGA 1/0795/16, KEGA 040ŽU-4/2018, KEGA
013ŽU-4/2019 and by the Slovak Research and De-
velopment Agency under the contract No. APVV-
0736-12 and No. APVV-14-0096.
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