Knot details name Boa knot image BoaKnot Final.jpg names type binding strength origin Peter Collingwood in 1996 related strangle knot , double constrictor knot releasing strength uses Best used for securing objects in cylindrical loads caveat abok number The boa knot is a modern, binding knot invented by weaving weaver Peter Collingwood in 1996. His intention was to develop a knot that would hold well when the constricted object was cut close to the winds of the knot. The boa knot is related to the strangle knot and the double constrictor knot . It combines both the structure and qualities of these other two knots. The boa knot is best used for securing objects in cylindrical loads. Image BoaKnot HowTo.jpg left thumb Boa knot step by step br style clear both See also List of binding knots List of knots External links http notableknotindex.webs.com boaconstrictor.html Knots knot stub ca Nus boa es Nudo boa ta ... more details
Knot details name Trilene knot image names type hitch strength 85 origin related releasing uses Attaching a fishing line to a hook or swivel caveat abok number The Trilene knot IPAc en icon t r a l i n is a multi purpose fishing knot that can be used for attaching monofilament line to hooks, swivels and lures. It resists slippage and failures. References refs External links http videofishingknots.com trilene knot.html Video instructions for tying the Trilene Knot http www.animatedknots.com trilene index.php Photographs on how to tie a trilene knot http dnr.state.il.us fish Knotgde.htm Drawings on how to tie a trilene knot http www.masterblanker.com articles trilene knot instructions Photographs showing how to tie a trilene knot Knots Category Fishing knots knot stub ... more details
Knot details name Handcuff knot image Handcuff knot ABOK 1140.jpg caption names Hobble knot type loop strength origin related Tom fool s knot , Fireman s chair knot releasing uses caveat abok number 412, 1134, 1140 , 2292 A handcuff knot is a knot tied in the bight having two adjustable loops in opposing directions, able to be tightened around hands or feet. The knot itself does not possess any inherent locking action, and thus is not as easy to use for such purposes as the name might suggest. The knot is also known as a hobble knot for similar reasons, from the idea that the knot was sometimes used on the legs of horses to limit the distance their riders had to walk in the morning to retrieve them. gallery Image Webeleinenstek 1.jpg 1. Two loops Image Fesselknoten 1.jpg 2. Pull through Image Fesselknoten.jpg 3. Tighten gallery The knot consists of two simple loops, overlaid, and with the ends pulled through. At that stage, the knot is slippery and easy to adjust. The knot can be locked by making one or more Overhand knot overhand knots with the loose ends in the manner of a Reef knot . ref name pawpock Des Pawson, Pocket Guide to Knots & Splices Edison, NJ Chartwell Books, Inc., 2002 , 146. ref The sizes of the two loops can also be fixed by making Half hitch es with each end over the necks of the loops. This configuration is known as the Fireman s chair knot . gallery Image Handcuff knot ABOK 1140 Reef finish.jpg Handcuff knot locked with an Overhand knot overhand knot Image Handcuff knot ABOK 1140 Hitch finish.jpg Handcuff knot locked with Half hitch half hitches , this is also known as the Fireman s chair knot gallery See also List of knots References reflist Knots de Fesselknoten fr N ud de menottes ... more details
Unreferenced auto yes date December 2009 Knot details name Bumper knot image Bumper knot diagram.png names Bait loop, Egg loop type loop releasing Non jamming uses fishing In fishing , a bumper knot also known as a bait loop or egg loop can be used to secure soft or loose bait, including clusters of eggs, to a hook. See also List of knots Knots DEFAULTSORT Bumper KnotKnot stub pt N de anzol ru ... more details
Knot details image name Stopper Knot names type strength origin related releasing strength uses Keeps the line from slipping out of things. The term stopper knot or simply stopper has three distinct meanings in the context of knotting and cordage . At the end of a line Image Ashley stopper lead ABOK 526.jpg thumb An Ashley stopper knot at the end of a line A stopper knot is tied at the end of a rope to prevent the end from unraveling, slipping through another knot, or passing back through a hole, Block sailing block , or Climbing equipment belay rappel device . Knots commonly used for this purpose are Overhand knot Double overhand knot Figure eight knot ropes Figure of eight knot Stevedore knot Ashley s stopper knot Around the standing part Image FlemishStopped.jpg thumb A Flemish bend stopped or backed up with a double fisherman s knot around each standing part When a stopper knot is tied outside another knot and around the standing part it can also be called a backup knot . Tying the end around the standing part helps prevent the knot from unraveling by not allowing the end to slide back into the knot. Examples of this usage are often seen in climbing , rope rescue , and other safety of life situations. Common knots used for this purpose are Overhand knot Double overhand knot strangle knot Double fisherman s knot Nautical usage In nautical settings, a stopper may refer to a length of rope that is belay ed at one end with the other end attached to a tensioned line using a friction hitch in order to slacken a portion of the tensioned line. For example if a Sheet sailing sheet becomes jammed on a winch while under sail, a stopper can be used to temporarily take the strain off the winch while the riding turn is cleared. Rolling hitch Other similar friction hitches See also Knot Reef knot References reflist Knots Category Knots Stopper Category Stopper knots ... more details
File BightLoopElbow.jpg thumb In reference to knot s, loop may refer to One of the fundamental structures used to tie knots. Specifically, it is a full circle formed by passing the working end over itself. A type of knot used to create a closed circle in a line. See also List of loop knots Knots DEFAULTSORT Loop Knot Category Knots Category Knot components ... more details
Knot details name Transom knot image Transom knot ABOK 1255.jpg caption names type lashing strength origin related Constrictor knot releasing Jamming uses Light duty right angle lashing caveat abok number 385, 1182, 1255 The Transom knot is a simple lashing knot , based on the constrictor knot , used to secure two linear objects at right angles to each other. To reinforce, a second Transom knot can be made on the opposite side and at a right angle to the first. Suggested for binding kite sticks. ref name ashley225 Clifford W. Ashley, The Ashley Book of Knots New York Doubleday, 1944 , 225. ref References references Knots knot stub ... more details
Infobox Heraldic knotknot Wake knot badge Wake badge.png badge caption The Wake badge heraldry badge . family Baron Wake of Liddell Wake family region Lincolnshire The Wake knot also known as an Ormonde or Ormond knot , a type of decorative knot , is a heraldic knot used primarily in English heraldry. It is most notable for its appearance on the Wake family badge heraldry heraldic badge , where it is formed by an intertwining rope and strap. This knot is similar in appearance to a Carrick bend . References reflist Heraldic knot Category Decorative knots Category Heraldic charges ... more details
Knot details image Diebesknoten 400px.jpg name Thief knot names Bag knot, Bread bag knot type binding origin Ancient related Reef knot , Granny knot , Grief knot releasing jamming, but not always strength caveat Knot Capsizing spills uses abok number 1207 The Thief knot resembles the reef knot except that the free, or working, ends are on opposite sides. It is said that sailors would secure their belongings in a WIKTIONARY ditty bag ditty bag using the thief knot, often with the ends hidden. If another sailor went through the bag, the odds were high the thief would tie the bag back using the more common reef knot, revealing the tampering, hence the name. It is difficult to tie by mistake, unlike the granny knot . The thief knot is much less secure than the already insecure reef knot. It unties itself if the lines are pulled when the same action would seize a reef knot. Image thief knot steps.svg thumb none Tying the thief knot step by step gallery Image Square knot.svg Reef knot Image Thief knot.svg Thief knot Image Granny knot.svg Granny knot Image Grief knot.svg Grief knot gallery See also List of binding knots List of knots External links http notableknotindex.webs.com reefknot.html The Reef Knot Family Knots de Diebesknoten eo Bopatrina nodo fr N ud de voleur nl Dievenknoop pl W ze z odziejski ru sv Tjuvknop vi N t n tr m ... more details
In the mathematics mathematical field of knot theory , a chiral knot is a knot mathematics knot that is not knot equivalence equivalent to its mirror image. An oriented knot that is equivalent to its mirror image is an amphichiral knot , also called an achiral knot or amphicheiral knot . The chirality of a knot is a knot invariant . A knot s chirality can be further classified depending on whether or not it is Invertible knot invertible . Background The chirality of certain knots was long suspected ... number knot theory crossing number , but a counterexample was found by Morwen Thistlethwaite ... of Knot Theory and Certain Applications of Knots and Links Bot generated title ref However, Tait s conjecture was proven true for prime knot prime , alternating knot s. ref name Mathworld align center class wikitable Number of knots of each type of chirality for each Crossing number knot theory ... 4 0 7 0 17 0 41 0 113 OEIS link A052400 Chiral knot gallery caption Both possible trefoil knot s. widths 80px heights 80px align right perrow 3 Image TrefoilKnot 02.png The left handed trefoil knot. Image TrefoilKnot 01.svg The right handed trefoil knot. gallery The simplest chiral knot is the trefoil knot , which was shown to be chiral by Max Dehn . All torus knot s are chiral. The Alexander polynomial cannot detect the chirality of a knot, but the Jones polynomial can in some cases if V sub k sub q V sub k sub q sup &minus 1 sup , then the knot is chiral, however the converse is not true. The HOMFLY polynomial is even better at detecting chirality, but no knot invariant is known which can fully detect chirality. ref http arxiv.org pdf hep th 9401095.pdf ref Reversible knot A chiral knot that is invertible knot invertible is classified as a reversible knot. ref name Atlas http katlas.math.toronto.edu wiki Three Dimensional Invariants Three Dimensional Invariants Knot Atlas Bot generated title ref Fully chiral knot If a knot is not equivalent to its invertible knot inverse or its ... more details
about a rope knot the mathematical version Granny knot mathematics Knot details name Granny knot image Granny knot.svg caption names False knot, Lubber s knot, Calf knot, Booby knot type binding strength origin related Reef knot , Thief knot , Grief knot releasing often jams uses caveat Should not be used as a bend. Inferior to reef knot for binding purposes, it can release suddenly and unpredictably. abok number 1206 The granny knot is a binding knot , used to secure a rope or line around an object. It is considered inferior to the reef knot , which it superficially resembles. Neither of these knots should be used as a bend knot bend for Bend knot attaching two ropes together . When attempting to tie a reef knot, it is easy to produce a granny knot accidentally. This is dangerous because the granny knot can slip when heavily loaded. A tightened granny knot can also jam and is often more difficult to untie than the reef knot. It is better to tie a reef knot in nearly all circumstances. One way to distinguish them is that in the reef knot each loop passes completely over, or completely under not through the neck of the other. The reef knot is commonly taught as left over right, tuck under then right over left, tuck under . The granny knot is the first step repeated twice, left over right, tuck under . This is a very common mistake made by people learning to tie a reef knot. Heraldry In heraldry, the granny knot is known as the Bourchier knot , due to being a heraldic badge of the John Bourchier, 1st Baron Berners Bourchier family . ref A Complete Guide to Heraldry by Arthur Charles Fox Davies 1909 , pp. 390, 469. ref Related knots gallery Image Square knot.svg Reef knot Reef square knot Image Thief knot.svg Thief knot Image Granny knot.svg Granny knot Image Grief knot.svg Grief knot gallery See also List of binding knots List of knots References references External links http notableknotindex.webs.com reefknot.html The Reef Knot Family Knots da K llingeknude de Altweiberknoten ... more details
distinguish fisherman s bend Knot details image Spierenstich.jpg name Fisherman s knot names Waterman s knot, Angler s knot, Englishman s knot type bend origin Ancient related Overhand knot , Double fisherman s knot , Triple fisherman s knot releasing Jamming strength caveat Difficult or impossible to untie uses Joining thin, stiff or slippery lines abok number 293, 1414 The fisherman s knot is a bend knot bend a knot for joining two lines with a symmetrical structure consisting of two overhand knot s, each tied around the standing part of the other. Other names for the fisherman s knot include angler s knot, English knot, halibut knot, waterman s knot. Though the fisherman s knot is associated with fishing, it can slip when tied in nylon monofilament and other slippery lines ref http www.marinews.com knots fishing knots Marinews Fishing Knots ref however, if more holding power is required, the overhand knots can be made with more turns, as in the double fisherman s knot , and so on. It is compact, jamming when tightened and the working end s can be cropped very close to the knot. It can also be easily tied with cold, wet hands. Though these properties are well suited to fishing , there are other knots which may provide superior performance, such as the blood knot . gallery Image Spierenstich 1.jpg 1. First overhand Image Spierenstich 2.jpg 2. Second overhand Image Spierenstich 3.jpg 3. Tighten gallery See also True lovers knot List of bend knots List of knots References reflist Knots Category Fishing knots ca Nus de pescador da Engelsk knob de Spierenstich es Nudo pescador fr N ud de p cheur it Nodo inglese he nl Vissersknoop ja pt N de pescador ru ta vi N t n i d y c u zh ... more details
Refimprove date October 2008 Howto date September 2009 Knot details name Diamond knot image Knife lanyard knot ABOK 787 Final.jpg caption names Knife lanyard knot, Sailor s knife lanyard knot, Marlingspike lanyard knot, Single strand diamond knot, Two strand diamond knot, Bosun s whistle knot, Friendship knot type loop strength origin related Carrick bend , Fiador knot releasing uses caveat abok number 787 , 2474 The diamond knot or knife lanyard knot is a knot for forming a decorative loop on the end of a cord such as on a lanyard . ref name ashley141 citation last Ashley first Clifford W. title The Ashley Book of Knots url accessdate origyear year 1944 publisher Doubleday location New York isbn page 141 pages quote ref Tying The diamond knot begins as a carrick bend with the ends exiting diagonally opposite each other. When the steps below are completed the knot is rearranged and tightened so that the ends emerge from the knot parallel and opposite their own standing part. gallery Image Knife lanyard knot ABOK 787 Carrick start.jpg Carrick bend start Image Knife lanyard knot ABOK 787 Over standing.jpg Working ends passed over each other s standing parts Image Knife lanyard knot ABOK 787 Through center.jpg The ends are passed up through the center of the carrick bend from below. gallery See also List of knots References reflist Knots DEFAULTSORT Diamond Knot Category Decorative knots Knot stub de Diamantknoten es Nudo diamante ... more details
Knot details name Arbor knot image Arbor knot diagram.png names type hitch strength origin related Honda knot releasing uses Fishing caveat abok number 1114 The Arbor knot is a typical fisherman fisher s knot . Its primary use is to attach fishing line to the arbor of a fishing reel . Tying an Arbor knot An arbor knot is tied by first passing the line around the reel arbor. The tag end is then tied in an overhand knot around the running line. Finally, an overhand knot is tied in the tag end. When tightened, the overhand knot in the tag end jams against the overhand knot tied around the running line. ref citebook title Fishing Basics The Complete Illustrated Guide author Gene Kugach year 1993 publisher Stackpole Books isbn 0811730018 ref See also List of knots References refs External links http www.fishing knots.com.au Arbor knot and other fishing knots http www.animatedknots.com arbor index.php Arbor knot at Animated Knots by Grog Knots Category Fishing knots knot stub de Arborknoten ta ... more details
Image Oriental tie knot.jpg thumb A small knot The small knot , or oriental knot or Kent knot , is the simplest method of tying a necktie , though some claim the simple knot is an alternate name for the four in hand knot . The small knot is not very well known despite its simplicity. Citation needed date May 2011 One of the reasons may be the fact that the small knot is not self releasing, and may annoy people accustomed to four in hand and Windsor knots who pull at the tie to untangle the knot. Additionally, some prefer that, if the thin end of the tie should become visible, that it not be inside out, and the small knot will be this way this can be remedied by twisting the tie one half turn while tying . Using the notation from The 85 Ways to Tie a Tie , the knot is tied Lo Ri Co T. gallery Image Tie diagram inside out start.svg Image Tie diagram inside out l r.svg Image Tie diagram inside out r c end.svg gallery See also Four in hand knot Pratt knot Half Windsor knot Windsor knot List of knots External links wikibooks How To Tie A Tie Dmoz Reference Knots Neckties Neckties Category Necktie knots fa it Nodo orientale pt N pequeno ta ... more details
Infobox Heraldic knotknot Shakespeare knot image image caption badge Shakespeare badge.png badge caption The Shakespeare badge heraldry badge . family Shakespeare disambiguation Shakespeare family region Warwickshire notes Named for William Shakespeare , who is believed to have borne the knot as an insignia. The Shakespeare knot , a type of decorative unknot , is a heraldic knot . It is a derivative of the Bowen knot and closely akin to the Dacre knot . The knot is most notable for its appearance on the Shakespeare badge heraldry badge . A signet ring preserved in Stratford upon Avon said to have belonged to William Shakespeare bears the knot, where its lower dexter and sinister loops entwine a W for William and an S for Shakespeare . ref cite book title Knots, Ties and Splices A Handbook for Seafarers, Travellers and All Who Use Cordage, with Historical, Heraldic and Practical Notes last Burgess first Joseph Tom authorlink coauthors year 1884 publisher G. Routledge and Sons location Oxford isbn pages p. 12 url ref References reflist Heraldic knot Category Decorative knots Category Heraldic charges ... more details
multiple issues unref April 2011 wikify April 2011 As a type of List of binding knots binding knot , the Jamming Knot is good for constricting a bundle of objects such as sticks or brush. It is basically a taut line hitch but the initial two wraps are on the outside of the working line rather than on the inside, and finished off with one wrap on the inside. Thus the knot holds tension towards the inside of the loop rather than the standing end of the rope as with the taut line turning a tension knot into a constricting knot. After the knot is tied, the knot is held with one hand, the standing end is pulled tight, and the knot should hold fast. This knot, as with the taut line, can be difficult or impossible to tie on slick or particularly stiff rope. References The author of this article found this knot on E How. Links to E How are blocked by Wikipedia. A simple search for adjustable jamming knot will find an article with step by step pictures. Knots Category Binding knots ... more details
Knot details name Manrope knot image Manrope knot.gif caption names type decorative strength origin related Turks head knot releasing uses caveat abok number A Manrope knot is a decorative kind of rope button. The original use of a manrope knot was to put at the end of the ropes on either side of a gangway leading onto a ship. This knot is often confused with a Turks head knot , as both knots have a basket weave pattern. External links http www.marinews.com bait presentation manrope knot 581 Knots Category Decorative knots knot stub de Fallreepsknoten nl Valreepsknoop ... more details
knot details name Snell knot image names type hitch strength origin related Knotless knot releasing uses Angling abok number The snell knot is a hitch knot used to attach an eyed fishing hook to fishing line . It passes through the eye of the hook, but primarily attaches to the shaft. It is a very secure knot, but because it is only easily tied using the near end as the working end, it is only used to attach a hook to a leader, rather than directly to the main line. Citation needed reason reliable source needed for the whole sentence date April 2012 Hooks can be bought pre snelled. External links http www.fintalk.com fishing knots common snell knot.html Illustrated instructions Knots DEFAULTSORT Snell Knot Category Fishing knots knot stub fishing stub ... more details
Image Blue 8 1 Knot.png thumb right A twist knot with six half twists. In knot theory , a branch of mathematics , a twist knot is a knot obtained by repeatedly twisting a closed loop topology loop and then linking the ends together. That is, a twist knot is any Whitehead double of an unknot. The twist knots are an infinite family of knots, and are considered the simplest type of knots after the torus knot s. Construction A twist knot is obtained by linking together the two ends of a twisted loop. Any number of half twists may be introduced into the loop before linking, resulting in an infinite ... Trefoil.png One half twist br trefoil knot Image Blue Figure Eight Knot.png Two half twists br Figure eight knot mathematics figure eight knot Image Blue Three Twist Knot.png Three half twists br Three twist knot 5 sub 2 sub knot Image Blue Stevedore Knot.png Four half twists br Stevedore knot mathematics stevedore knot Image Blue 7 2 Knot.png Five half twists br 7 sub 2 sub knot Image Blue 8 1 Knot.png Six half twists br 8 sub 1 sub knot gallery Properties All twist knots have unknotting number one, since the knot can be untied by unlinking the two ends. Every twist knot is also a 2 bridge knot . ref cite book author Rolfsen, Dale title Knots and links publisher AMS Chelsea Pub location Providence ... the unknot and the stevedore knot mathematics stevedore knot are slice knot s. ref MathWorld title Twist Knot urlname TwistKnot ref A twist knot with math n math half twists has crossing number knot theory crossing number math n 2 math . All twist knots are invertible knot invertible , but the only amphichiral knot amphichiral twist knots are the unknot and the figure eight knot mathematics figure eight knot . Invariants The invariants of a twist knot depend on the number math n math of half twists. The Alexander polynomial of a twist knot is given by the formula math Delta t begin cases frac ... V q frac q 3 q q 3 n q n q 1 . math References reflist Category Knot theory ... more details
about a rope knot the mathematical version Stevedore knot mathematics Knot details name Stevedore knot image Double eight 2.JPG names Double figure eight type stopper strength origin related Figure eight knot ropes Figure of eight knot , Overhand knot , Figure of nine loop , Ashley s stopper knot releasing Non jamming uses To provide a bulky, secure when slack stopper caveat abok number 456, 522 The Stevedore knot is a Stopper knot stopper knot , often tied near the end of a rope . It is more bulky and less prone to jamming than the closely related Figure eight knot ropes figure of eight knot . Naming There is a lack of consensus among knot experts regarding the origin of the name. Many sources, including The Ashley Book of Knots , suggest the knot was used by stevedore s in their work loading and unloading ship s. To raise and lower cargo they used large block sailing blocks and these required a larger stopper knot to prevent the line from running completely through the block. ref name ashley85 Clifford W. Ashley, The Ashley Book of Knots New York Doubleday, 1944 , 85. ref However, in The Art of Knotting & Splicing , Cyrus Day disagrees, stating the name originated in a pamphlet issued about 1890 by the C.W. Hunt Company, which sold rope under the name Stevedore. It was subsequently adopted by dictionaries, engineers handbooks, and other works of reference, and it is now firmly established in books, if not in the vocabulary of seamen. ref name aksday Cyrus Lawrence Day, The Art of Knotting and Splicing, 4th ed. Annapolis Naval Institute Press, 1986 , 40. ref Tying Image Double eight 1.JPG thumb Stevedore knot before tightening The knot is formed by following the steps to make a figure of eight knot, but the working end makes an additional wrap around the standing part before passing back through the initial loop in the same direction it would have for a figure of eight knot. See also List of knots References references External links http mathworld.wolfram.com StevedoresKnot.html ... more details
Image Blue Cinquefoil Knot.png thumb right A cinquefoil knot. In knot theory , the cinquefoil knot , also known as Solomon s seal knot or the pentafoil knot , is one of two knots with crossing number knot theory crossing number five, the other being the three twist knot . It is listed as the 5 sub 1 sub knot in the Alexander Briggs notation , and can also be described as the 5,2 torus knot . The cinquefoil is the closed version of the double overhand knot . The cinquefoil is a prime knot . Its writhe is 5, and it is invertible knot invertible but not amphichiral knot amphichiral . ref MathWorld title Solomon s Seal Knot urlname SolomonsSealKnot ref Its Alexander polynomial is math Delta t t 2 t 1 t 1 t 2 math , its Conway polynomial is math nabla z z 4 3z 2 1 math , and its Jones polynomial is math V q q 2 q 4 q 5 q 6 q 7 . math ref http katlas.math.toronto.edu wiki 5 1 5 1 Knot Atlas Bot generated title ref Surprisingly, these are the same as the Alexander, Conway, and Jones polynomials of the knot 10 sub 132 sub . However, the Kauffman polynomial can be used to distinguish between these two knots. The name &ldquo cinquefoil&rdquo comes from the five petaled flowers of plants in the genus Potentilla . File Cinquefoil Knot.jpg right thumb Edible cinquefoil knot. See also Pentagram Trefoil knot 7 knot Skein relation References reflist Category Knot theory knottheory stub ... more details
File Hastings badge.png thumb right 200px The heraldic badge of the Hastings family, with the so called Hastings knot entwining a sickle and garb of wheat. The Hungerford or Hastings knot is a type of decorative unknot , and is a heraldic knot used primarily in English heraldry. It is most notable for its appearance on the Hungerford and Hastings family badge heraldry heraldic badge s. References reflist Heraldic knot Knots Category Decorative knots Category Heraldic charges knot stub heraldry stub ... more details
File Lacy knot.png thumb right 200px The Lacy knot. The Lacy knot is a decorative heraldic knot , ref Charles Boutell, English heraldry. With 450 illus. drawn and engraved on wood by R.B. Utting , Cassell, Petter & Galpin, 1867, p. 135 ref the badge heraldry badge of the De Lacy family. It features in the Crest heraldry crest of the borough of Burnley borough Burnley . References reflist Knots knot stub heraldry stub Heraldic knot Category Decorative knots Category Heraldic charges ... more details
In mathematics , a knot mathematics knot is an embedding of a circle into 3 dimensional Euclidean space . The knot group of a knot K is defined as the fundamental group of the knot complement of K in R sup 3 sup , math pi 1 mathbb R 3 backslash K . math Other conventions consider knots to be embedded in the 3 sphere, in which case the knot group is the fundamental group of its complement in S sup 3 sup Properties Two equivalent knots have isomorphic knot groups, so the knot group is a knot invariant and can be used to distinguish between inequivalent knots. This is because an equivalence between two knots is a self homeomorphism of math mathbb R 3 math isotopic to the identity sending the first knot onto the second. Clearly the homeomorphism restricts onto a homeomorphism of the complement of the knot therefore inducing an isomorphism in the fundamental group of the knot complement. However, two knots can have isomorphic knot groups without necessarily being equivalent see below for an example . The abelianization of a knot group is always isomorphic to the infinite cyclic group Z this follows because the abelianization agrees with the first homology group , which can be easily computed. The knot group or fundamental group of an oriented link in general can be computed in the Wirtinger presentation by a relatively simple algorithm. Examples The unknot has knot group isomorphic to Z . The trefoil knot has knot group isomorphic to the braid group B sub 3 sub . This group has the presentation ... aba bab rangle math . A p , q torus knot has knot group with presentation math langle x,y mid x p y q rangle math . The Figure eight knot mathematics figure eight knot has knot group with presentation math langle x,y mid x 1 yxy 1 xy yx 1 yx rangle math The square knot mathematics square knot and the granny knot mathematics granny knot have isomorphic knot groups, yet these two knots are inequivalent. See also Link group References Springer id k055580 title Knot and link groups Category Knot theory ... more details
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