Knots

Nomenclature in the world of knots is inconsistent in any language.  Within English some would stipulate that the tangles of cordage we commonly call knots should actually refer to only those things that are neither bends nor hitches.   Ideally a bend should join two ropes or lines together, whereas a hitch should attach a line to a post, ring, rail or something.  In general however, the term knot is used to encompass all three.

Some fundamental knot component terms include “working or tag end”, “standing line”, bight and loop.  In a bight the end and the standing line are parallel but in a loop the working end crosses over the standing part.  Other knot terminology might include: braids, bindings, coils, dog, elbow, friction hitch, lashing, lanyard, locking tuck, messenger, nip, noose, round turn, plait, seizing, sling, splice, stopper, trick or whipping.  A knot that has a draw loop is said to be a slipped knot, which is not the same thing as a proper slip knot.  When tying shoelaces for example two draw loops or bights finish the knot and provide easy untying.

The simplest knot of all is the “Overhand knot”.  Once tied in a line of rope or cordage, every knot reduces the static tensile strength or average breaking strength of that line, when tension is applied.  The proportion of knotted cordage’s breaking strength relative to its unknotted strength describes a given knot’s “efficiency“.  Efficiency is about the only common, measurable, descriptive term shared between knots, bends and hitches.  Most knots have an efficiency between 40% and 80%.  The overhand knot (ABoK#514) has an efficiency rating of 50%, which is poor because when stressed it reduces the strength of a line by half.

Several knots we are familiar with are ancient.  Long ago prehistoric fishermen were using knots to make gill, casting and trawling nets. In addition to practical knots, the ancient Tibetans, Chinese and Celts contemplated some very intricate and elaborate decorative knots.

There is by no means an authoritative categorization or listing of all knots.  Growing in acceptance, the closest thing to an authoritative list of working knots might be Clifford W. Ashley’s illustrated encyclopedia of knots.   First published in 1944, The Ashley Book of Knots list and numbers more than 3,800 basic knots, but this does not even come close to enumerating all the variants and ornamentals in existence.  There is a lively online forum on almost every subject related to knots – hosted by the International Guild of Knot Tyers.  Also there is a quick and handy online knot index which features images for some of the more common working knots.

* A tangential detour: Knot Theory

Lest the reader assume that knots are an overly simplistic or entirely trivial subject they should realize that the future advancement of computing may rely upon an underlying study of knots.  The speed of the fastest computers is approaching a limit due to the finite speed of the electron itself.  Any increased computing speed in the future may depend upon quantum field theory and statistical mechanics; mathematics that sprouted from a topology known as “knot theory” or the mathematical study of knots.  Knot theory is often applied in geometry, physics and chemistry. Topology is concerned with those properties that don’t change when an object is continuously stretched, twisted or deformed.  Topology involves set theory, geometry, dimension, space and transformation.  Topology studies spatial objects (objects that occupy space), the space-time of general relativity, knots, fractals and manifolds.  A mathematical knot is one where the ends are joined together to prevent it from becoming undone.  Inspired by real world knots, the founders of knot theory were concerned with knot description and complexity.  They created tables of knots and links (knots of several components entangled together).  Over 6,000,000,000 knots have been tabulated to date and obviously concise tabulation would be a task for a machine and not a human.

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A FEW GOOD KNOTS

A surprising number of people are unfamiliar with or cannot tie a decent knot, when such a skill can occasionally prove to be quite handy.  A repertoire of only a dozen or so well chosen knots will stand the survivalist or Boy Scout in good stead with his contemporaries.  An effective working knot should have practical applications, it should be simple to tie and easy to remember and in most instances it should be easy to untie.  My subjective list of six of the most important and effective working knots include the slipped -slipknot, bowline, figure -8 (or Figure of Eight Loop), clove hitch, prusik knot and the trucker’s hitch.   The clove hitch and prusik knots are fundamental in that several useful variations have been built upon them.

The simple slipknot tightens as the hauling end is pulled and can become very tight and difficult to untie.  By “slipping” the knot with a bight or draw loop however, even the tightened knot will fall apart after a stout yank of the tag end.  This simple knot is appropriate in many applications including tying a hammock to a tree or fastening a horse halter to a post or rail so that it can be unfastened quickly in an emergency.

Many knots including the venerable bowline can be “slipped” in such a fashion.  For those people who encounter a mental block when trying to remember how to tie a bowline, there is an easily remembered right-hand–twist method to use.

There are many instances when a loop in the middle of a line is called for.  As an example, for safety a mountain climber might tie himself to a middleman’s knot in the center of a climbing rope.  While a simple overhand loop might suffice in this application – it could become difficult to untie after being stressed.  The addition of another twist to the overhand loop results in the so-called Figure of Eight loop which is probably more efficient and much easier to untie.  Some might consider the Figure of Eight loop (or Flemish loop) preferable to comparable mountaineering knots like the Alpine Butterfly, merely because it is simpler and easier to remember.

The granddaddy of all “ascending knots” or “friction hitches” is the venerable Prusik knot which was first created during WWI and named for its inventor.  The Prusik can be doubled (with 6 coils rather than 4) to produce more traction.  The younger Kleimheist also shown in the illustration below is also popular with modern day climbers.

Few good (simple) ascending knots for mountaineering can be tied with nylon webbing.  The Heddon and double Heddon knots shown next are exceptions that seem appropriate.

The Trucker’s hitch is an important and utilitarian cinching knot that is actually a compound construction of two other knots.  Disregarding friction, the Trucker’s hitch can tightly strap down loads on trucks, trailers, boats and pack saddles because it applies a 2:1 mechanical advantage.  The standing line employs a ring, carabineer or middleman’s loop while the cinch is tightened with the tag end.  After the cinch is drawn tight the pressure is held by pinching the bight with one hand, before finishing with a simple slipped overhand knot.

The finial knot (of the six most crucial selected here) is the excellent, general purpose ‘clove hitch’.  It is mentioned last because many admirable variations have been conceived from it, and illustrations of a few of those will follow.

Excellent for sacks and trash bags the “constrictor knot’ differs only slightly from the clove hitch, but holds more firmly.  It can be hard to untie unless intentionally slipped with a draw loop.

When wrapped around a tent stake the “taut line hitch” below is useful for tensioning a tent guy line.  To the right of that is a useful clove hitch variant that has no recognized common name or ABoK number.  Tentatively referred to as the wireline hitch here, the grip of this variant is superior to the taut line version.

 

A few more knots _ deserving honorable mention

Strong and efficient the ‘Palomar knot’ is useful for attaching large hooks, lures or sinkers to a fishing line.

The “Surgeon’s loop” is another simple and effective knot for attaching small lures or flies to a tiny mono-filament fishing line.  Knots like the surgeon and Palomar are cut away rather than untied after they serve their purpose.

The “Ossel hitch” is an ancient knot; no one knows how old. It is or was a simple, secure and effective knot used to suspend gill nets from a larger line.  Strangely the ossel hitch is not recognized in Ashley’s encyclopedia.  This may be because “ossel” is a Scottish word and was not that familiar when Ashley illustrated his book.  There is a similar but different knot in the encyclopedia known as the “Netline Knot” (ABoK #273) that hails from Cornwall on the southern coast of England.

This simple Anchor Bend variant below is easily remembered and is much more secure than the parent knot.

Finally, this old page construction below introduces a couple of utilitarian gripping hitches

 

KNOT CONUNDRUMS

This is a blog post and not an encyclopedia therefore most knots cannot be shown.  Returning to the off topic tangent of knot mathematics we come to a group of abstract ideas known as graph theory which foreshadowed or laid the foundation for topology.  The father of graph theory was a Swiss mathematician and physicist named Leonhard Euler.   Leonhard discussed a notable historical problem in mathematics called “The Seven Bridges of Konigsberg”.  The unsolvable problem was to walk through the city, crossing each bridge once and only once.  What is called Euler’s solution does not solve the impossible but became the first theorem of planar graph theory. <new link May 2019>

 

* Back in 1735 the seven bridges of Konigsberg were real and that city was part of the Prussian Empire and bordered Poland on the Baltic. Konigsberg, Prussia became Kaliningrad, Russia (54°42’12” N, 20°30’56”E) sometime after WWI. After the breakup of the Soviet Union, Kaliningrad and surrounding province became physically separated from the rest of Russia. After another world war and the ravages of time only two of the original bridges from Euler’s time survive. Five bridges now connect the city and islands formed by the Pregel River.

A similar conundrum that Euler might have considered had he the chance is the hypothetical house with five rooms and sixteen doors. The object is for a person to walk through each door once, but one time only.

Finally we come to the perplexing Mobius strip and Trefoil knot. The naughty Mobius strip is something of a paradox. The single edge of a Mobius strip is topologically equivalent to the circle and mathematically it is non-orientable.

A physical Mobius strip can be constructed from a belt or strip of paper.  One simply grabs the two ends and gives one end a half twist before taping the two together in a loop.  The resulting surface then has only one side and one edge.  Imagine a miniature gravity defying car driving around the surface of the strip.  If the car began on the top side of the surface then its path after one revolution of the loop would place it on the bottom side of the surface.  Consider a bug dragging a paintbrush while walking along the right edge of the strip and making two revolutions of the loop.  We perceive two edges to the strip but realize there is only one.

M.C. Escher incorporated the Mobius strip in some of his graphical art.  In the real world recording tapes and typewriter ribbons have been spliced in the continuous-loop – Mobous strip fashion to double playing time or ink capacity.  Large conveyor belts have also been wrapped the same way, to increase belt life by doubling the surface area.  The Mobius strip has several curious properties.  A continuous line drawn down the middle of the loop will be twice as long as the same loop.  Cutting this paper loop down the centerline will produce one long loop with two twists (not two strips) and finally two edges.  Cutting this longer strip again as before, will produce two strips, each with two full twists and intertwined together.

In topology the “unknot” is a circle and the “trefoil knot” is the simplest knot. Named after the plant that produces the three-leaf clover, the trefoil knot can be tied by joining together the two loose ends of a common overhand knot, but this results in a knotted loop.  Although it doesn’t look very convincing when done with paper, a trefoil knot can also be constructed by giving a band of paper three half twist before taping the ends and then dividing it lengthwise.