Definitions

# knot theory

Mathematical theory of closed curves in three-dimensional space. The number of times and the manner in which a curve crosses itself distinguish different knots. The fewest possible crossings is three, for the overhand (trefoil) knot, which occurs in two mirror versions according to the directions in which the curve crosses itself. Knot theory has been used to understand both atomic and molecular structures (protein folding).

Examples of common knots.

In cording, the interlacement of parts of one or more ropes, cords, or other pliable materials, commonly used to bind objects together. Knots have existed from the time humans first used vines and cordlike fibers to bind stone heads to wood in primitive axes, and were also used in the making of nets and traps. Knot making became sophisticated when it began to be used in the ropes, or rigging, that controlled the sails of early sailing vessels, and thus became the province of sailors. Knots are still depended on by campers and hikers, mountaineers, fishermen, and weavers, among others.

A knot is a method for fastening or securing linear material such as rope by tying or interweaving. It may consist of a length of one or more segments of rope, string, webbing, twine, strap, or even chain interwoven such that the line can bind to itself or to some other object—the "load". Knots have been the subject of interest for their ancient origins, their common uses, and the mathematical implications of knot theory.

## Usage

There are a large variety of knots and each knot has specific properties and suitability for a range of tasks. Some knots are well-adapted to attach to particular objects such as another rope, cleat, ring, or stake. Other knots are made to bind or constrict around an object. Decorative knots usually bind to themselves to produce attractive patterns. Choosing the correct knot for the job at hand is one of the most fundamental aspects of using knots well. However, if memory is limited, three of the most useful knots are the bowline, the sheet bend, and the clove hitch.

### Learning

The number of books, websites, videos, and other resources available to those interested in learning about knots is a testament to the value they hold for humankind. While some people possess an innate ability to look at a diagram or photo and tie the illustrated knot, for others the initial stages of learning are best accomplished by being shown knot tying methods by a person who already knows them. Knot tying skills are often transmitted by sailors, scouts, climbers, cavers, arborists, rescue professionals, fishermen, and surgeons. After mastering a few basic knots, the diagrams and photos become easier to interpret and use to continue the learning process. As more knots are learned, patterns begin to become evident in their structure and methods of tying. The learning of knots rewards practice and patience.

### Applications

Knots are essential in many industrial, occupational, recreational, and domestic settings. Even simple activities such as running a load from the hardware store to home can result in disaster if a clumsy twist in a cord passes for a knot. Truckers needing to tie down a load may use a trucker's hitch, gaining mechanical advantage. Knots can save the spelunker from foolishly becoming buried under millions of tons of rock. Whatever the activity, such as sailing on the water or climbing on a cliff-side rock, learning well-tested knots prior to some hazardous activity introduces a critical measure of safety. In addition to safety, appropriate knots can prevent the necessity of cutting lines.

Knots can be applied in combination to produce complex objects such as lanyards and netting. In ropework, the frayed end of a rope is held together by a type of knot called a whipping knot. Many types of textiles use knots to repair damage. Macrame, one kind of textile, is generated exclusively through the use of knotting, instead of knits, crochets, weaves or felting. Macramé can produce self-supporting three dimensional textile structures, as well as flat work, and is often used ornamentally or decoratively.

### Basic useful knots

Some of the most useful everyday knots are the following. Most are both secure and easy to untie:

• For tying a loop in the end of a rope, as around your waist or to secure a ring or grommet: the bowline.
• For tying the ends of two ropes together: the sheet bend works well even with two different ropes. This is just as easy to tie but much more secure than the square knot, which is not an effective knot for joining two freestanding ropes. If one rope is much thicker than the other, a double sheet bend is better. (Materials such as cables which are not easily tied may sometimes be joined by two interlocking bowlines instead.)
• The fisherman's knot is stronger but suffers the cost of being difficult to untie once loaded.
• For flat material such as (seat)belts,and webbing, the water knot is best. (However, this is a poor knot for tying ropes.)
• For tying a rope to a pole, the buntline hitch. A slipped variant is useful for quick release. If both ends of the line will be loaded, then the clove hitch will suffice.
• On square posts where the clove hitch is not secure, two half-hitches is good. It can be hard to untie, so the slipped variant may be useful.
• The timber hitch works well on rough surfaces, including square timber, especially under constant strain, but isn't secure if the load jumps around. The Killick hitch is a variant used for hoisting rocks and other odd shapes.
• The rolling hitch is useful when you don't want a rope to slide up on down a pole, or when tying one rope to the middle of another.
• For a clothesline or other line that sags over time, the taut-line hitch can be ratcheted up to take out the slack.
• The trucker's hitch is useful for clinching down a load.
• The sheep shank is useful for taking slack out of the middle of a rope, but will only hold as long as there's strain on the rope.
• The constrictor knot works well for making bundles or tying the neck of a sack. However, it is nearly impossible to untie once tightened, and will likely need to be cut with a knife.
• The alpine butterfly puts a secure loop in the middle of a rope when the ends aren't free.
• For climbing a rope, the Prusik knot allows you to make footholds out of loops of narrower rope which ratchet up the main rope. (The loop can be made by tying together the ends of a rope with a double fisherman's knot.)
• The diamond hitch works well for packing trail animals.
• The figure-of-eight knot stops the end of a rope from slipping through a hole or other tight spot.

## Properties

### Strength

Knots invariably weaken the rope they are made in. When knotted rope is strained to its breaking point, it almost always fails in or near the knot, unless it is defective or damaged elsewhere. The bending, crushing, and chafing forces that hold a knot in place also unevenly stress the rope fibers and ultimately lead to the reduction of strength. The exact mechanisms that cause the weakening and failure are complex and are the subject of continued study.

The relative knot strength, also called knot efficiency, is the breaking strength of a knotted rope as a proportion of the breaking strength of the rope without the knot. There are many difficulties in determining an overall numeric knot efficiency for a given knot. This is due to the many factors that can affect the results of a knot efficiency test: the type of fiber, the style of rope, the size of rope, whether it is wet or dry, how the knot is dressed before loading, how rapidly the knot is loaded, whether the knot is repeatedly loaded, and so on. With those limitations noted, most common knots have an efficiency between 40% and 80%.

While some rope splices can retain nearly the full strength of the rope when forming loops and bends, conventional knots are much more practical in most situations. Thus the prudent knot user will always allow for a large safety margin in the strength of rope chosen for a task due to the weakening effects of knots, aging, damage, shock loading, etc. In general, the safe working load is often specified as between 10% and 20% of the rated breaking strength of the rope being used. For safety of life applications many other factors come into play which are beyond the current scope of this article. Experienced practitioners should always be consulted before using ropes and knots when safety of life, limb, or property is involved.

### Security

Even if the rope does not break, a knot may still fail to hold. A knot which holds firm under a variety of adverse conditions is said to be more secure than one that does not. The main ways knots fail to hold are:

#### Slipping

The tension from the load causes the rope to work back through the knot in the direction of the load. If this continues far enough, the working end will pass into the knot and the knot unravels and fails. This behavior can be worsened when the knot is repeatedly strained and let slack, dragged over rough terrain, or repeatedly impacted such as against a mast or flagpole.

Even with secure knots, some slippage may occur as the knot is first put under real tension. This can be dealt with by leaving plenty of rope at the working end outside of the knot and by dressing the knot cleanly and tightening it as fully as possible before loading. In some cases the use of a stopper knot or, even better, a backup knot can prevent the working end from passing through the knot, but it is generally better to use a more secure knot if one is observed to slip. In life critical uses backup knots are often added to already secure knots in order to maximize safety.

#### Capsizing

Capsizing (or spilling) a knot is changing its form, rearranging its parts, usually by pulling on specific ends in specific ways. Some knots when used in an inappropriate way tend to capsize easily or even spontaneously. Often the capsized form of the knot offers little resistance to slipping or unraveling. For an excellent example of a knot that capsizes dangerously, see the discussion of the reef knot used as a bend.

Sometimes a knot is intentionally capsized as a method of tying another knot, such as the "lightning method" of tying a Bowline. Some knots, such as the Carrick Bend, are generally tied in one form and then capsized to attain a stronger or more stable form.

#### Sliding

In knots that are meant to grip another object, failure can be defined as the knot moving relative to the object being gripped. While the knot itself does not fail, it ceases to perform the desired function. For example a simple Rolling Hitch tied around a railing and pulled parallel to the railing might hold to a certain tension and then start sliding. Sometimes this can be corrected by working-up the knot tighter before subjecting it to load but usually a knot with more wraps, or a different size or type of rope will need to be used.

## Components

Bight

The center part of a length of rope, string, or yarn as opposed to the ends.
*A "bight" is any curved section, slack part, or loop between the ends of a rope.
*The phrase "in the bight" implies a U-shaped section of rope is itself being used in making a knot. Many knots can be tied either with the end or in the bight. Bitter end
More a ropeworker's term than a knot term, the reference is to the end of a rope that is tied off, hence the expression "to the bitter end". A bitt is a metal block with a crosspin used for tying lines to, found on docks. In fact the bitter end is the end of the Anchor "Cable" that connects to the Anchor Bitts in the cable locker under the forecastle or poop using the bitter pin. (British nautical usage). Other uses are borrowed from this derivation.Loop :A full circle formed by passing the working end over itself. Note that the term 'loop' is also used to refer to a category of knots (see 'Categories' below).Elbow :Two crossing points created by an extra twist in a loop.Standing end
The end of the rope not involved in making the knot, often shown as unfinished.Standing part
Section of line between knot and the standing end.Turn
*A turn or single turn is a single pass behind or through an object.
*A round turn is the complete encirclement of an object; requires two passes.
*Two round turns circles the object twice; requires three passes.Working end
The active end of a line used in making the knot. May also be called the 'running end' or 'live end'.Working part
Section of line between knot and the working end.

## Categories

The list of knots is extensive, but common properties allow for a useful system of categorization. For example, loop knots share the attribute of having some kind of an anchor point constructed on the standing end (such as a loop or overhand knot) into which the working end is easily hitched to using a round turn. An example of this is the bowline. Constricting knots often rely on friction to cinch down tight on loose bundles; an example is the Miller's knot. Knots may belong to more than one category.Bend :A knot uniting two lines (for knots joining two ends of the same line, see binding knots or loops). List of bends.Binding :A knot that restricts object(s) by making multiple winds. List of binding knots.Coil :Knots used to tie up lines for storage. List of coil knots. Another handy coil is Aufschießen einer Leine.Decorative knot :A complex knot exhibiting repeating patterns often constructed around and enhancing an object. List of decorative knots.Hitch :A knot tied to a post, cable, ring, or spar. List of hitch knots.Lashing :A knot used to hold (usually) poles together. List of lashing knots.Loop :A knot used to create a closed circle in a line. List of loop knots.Plait (or Braid):A number of lines interwoven in a simple regular pattern. List of plait knots.Slip (or Running) :A knot tied with a hitch around one of its parts, contrasted. with a loop, which is closed with a bend. A slip knot can be closed, a loop remains the same size. List of slip knots. Seizing :A knot used to hold two lines or two parts of the same line together. List of seizing knots.Sennit :A number of lines interwoven in a complex pattern. List of sennit knots.Splice :A knot formed by interweaving strands of rope rather than whole lines. More time consuming but usually stronger than simple knots. List of splices.Stopper :A knot tied to hold a line through a hole. List of stopper knots.Trick :A knot that is used as part of a magic trick, a joke, or a puzzle. List of trick knots.Whipping :A binding knot used to prevent another line from fraying.

## Knot theory

Knot theory is a branch of topology, and deals with the mathematical analysis of knots, their structure and properties, and the relationships between different knots. In topology, a knot is a figure consisting of a single loop, abstracted from any physical rope or line, with any number of crossing or "knotted" elements. As such, it has no proper ends, and cannot be undone or untied. Various mathematical techniques are used to classify and distinguish knots, for example the Alexander polynomial can be used to distinguish the trefoil knot from the figure-of-eight knot and the unknot (a simple loop).

## References

• Clifford W. Ashley The Ashley Book of Knots. Doubleday, New York. ISBN 0-385-04025-3
• R.S. Lee. All The Knots You Need. Algrove Publishing. ISBN 0-921335-47-4
• Raoul Graumont. Handbook of Knots. Cornell Maritime Press/Tidewater Publishers. ISBN 0-87033-030-6
• Cyrus L. Day. Knots & Splices. International Marine/McGraw-Hill Companies. ISBN 0-87742-252-4
• Geoffrey Budworth (1999). The Ultimate Encyclopedia of Knots & Ropework. Annes Publishing Limited. ISBN 1-55267-986-1
• John Cassidy (1985). The Klutz Book of Knots. Klutz Press, Palo Alto, California. ISBN 0-932592-10-4
• Des Pawson(2001). Pocket Guide to Knots & Splices. Produced for Propsero Books by RPC Publishing Ltd., London. ISBN 1-55267-218-2
• Brion Toss. The Complete Rigger's Apprentice. International Marine/McGraw-Hill Companies. ISBN 0-07-064840-9
• Allen Padgett and Bruce Smith. On Rope. National Speleological Society. ISBN 0-9615093-2-5
• J.C. Turner and P. van de Griend (ed.) (1996). History and Science of Knots. World Scientific. ISBN 981-02-2469-9