The Knots in the Quipu, and in the Friar's Belt
In a canonical numerical quipu, each pendent (or subsidiary) displays a number: a positive integer, expressed in decimal notation...
Tony Phillips
Stony Brook University
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Introduction

Quipus or khipus are a recording medium developed by the Inca and their predecessors (our earliest examples date back to about 900 AD; many details of their use were unfortunately obliterated during and after the Conquest, in the early 16th century; there are about 600 quipus [Urton] known to exist today, mostly in museums). Signs in this medium are knots in strings.
This illustration, from the cover of [Locke], shows the quipu longknot representing the number 3.

Quipus and their knots
Left: a numerical quipu in the collection of the American Museum of Natural History, New York (Accession No. B8713). In this example the pendents have no subsidiaries, but six top cords are attached so as to group the pendants four by four. As Locke's analysis shows, each top cord records the sum of the numbers encoded in knots on the four pendents in its group. This item, which Locke describes as "an example of the most highly developed form of the quipu," appears as the frontispiece to his work [Locke]. An even more intricate numerical relation, this time between different quipus, all found together, has been recently documented in [Urton and Brezine].
Right: Locke does not give the dimensions of that quipu, but they may be guessed by comparison with the quipu shown (with some of its knots visible) in this page from El primer nueva corónica y buen gobierno, by Felipe Guaman Poma de Ayala (1615). Image from the Guaman Poma Website, Royal Library, Copenhagen, Denmark, used under Creative Commons License. Caption: Storehouses of the Inca. We see an administrador, identified as Poma Chaua (an ancestor of the author) showing a quipu, presumably with some tally of the storehouses' contents, to Topa Inca Yupanqui (147193). Text is in Spanish and Quechua.
A quipu is a portable device made of (cotton or camelid wool) string in a twodimensional array. A primary cord (also, main cord, transverse cord) "varying in length from a few centimeters to a meter or more" [Locke] supports several pendents (also, pendant cords) "seldom exceeding 0.5 meter in length" [Locke]; up to 1500 in the largest examples, but typically in the range from a few to several hundred. The pendents can bear subsidiary cords, which themselves can have subsidiaries, and so on, up to six levels in some instances [Ascher]. There may also be a set of top cords, attached so as to lie most naturally on the opposite side from the pendants. The pendents, subsidiaries and top cords each carry a sequence of knots, which record information.
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The knots
Locke's portrayal of primary cord, pendents and susidiaries, and of the three types of informationbearing knots that appear in standard ancient quipus: the overhand knot (i, j, 1_{j}), the figureeight knot (g, h, 1_{h}), and the long knots tied as an overhand knot but casting the end through the loop from 2 to 9 times (2, ..., 9; 4 is shown loosened as e, and tied in the subsidiary as l; 3 also appears on a pendent as f, and 5 appears on pendent d which, without its subsidiary, encodes the number 135. As remarked in [Urton], each of these knots can be tied either righthand or lefthand.
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The grammar of knots
In a canonical numerical quipu, such as Locke's example shown above, each pendent (or subsidiary) displays a number: a positive integer, expressed in decimal notation, as follows:

units appear lowest on the cord. 1 is represented by a figureeight knot, 29 by the corresponding long knot.

tens appear one level higher. 10 is represented by a single overhand knot on that level, 20 by a cluster of two overhand knots, up to 90: a row of nine closely spaced overhand knots.

hundreds appear one level above the tens, and are similarly represented: 100: a single overhand knot, up to 900: a row of nine overhand knots.

similarly for higher powers of ten. The Khipu Database website, based at Harvard, shows an example, also from the AMNH but presumably acquired after Locke's study, where six positions are used.

A zero in any place corresponds to an unknotted stretch of cord at that level.
The different levels are apparent in Locke's photograph, and even more precisely demarcated in the sixfigure example linked to just above. As in Old Babylonian notation, the lack of a symbol for zero introduces a theoretically possible ambiguity: 210 and 2100 alone cannot be distinguished. But if just one cord in a quipu displays a number with a nonzero units place, then the distinctive knots used for that position, and the coherence of levels, will anchor the entire array. The ideal even spacing between levels shown in the sixfigure example could compensate the (unlikely) absence of nonzero entries, in any position higher than units, across an entire quipu.
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From a topological point of view
To a topologist, the knots in quipus are quite standard. When the two free ends of the overhand knot are joined the configuration forms the trefoil knot, knot $3_1$ in the standard knot table. Similarly the figureeight knot appears as $4_1$. The first few long knots are in the table: the quipu knot 2 is $5_1$, 3 gives $7_1$ and 4 gives $9_1$. There are 165 distinct knots with 10 crossings; the table stops there. But if it went on, presumably the quipu 5 would correspond to $11_1$, ..., and 9 to $19_1$ since these torus knots are the simplest in each case.
Locke's loosened 4knot, captured in a loop, can be deformed to the standard picture of $9_1$.
Each of the torus knots $3_1$, ..., $19_1$ exists in two mirrorimage forms, distinct in that one cannot be deformed to the other. All 18 of these forms exist in the corpus of ancient quipu knots ([Urton], Chapter 3).
The figureeight knot also exists in quipus in two mirrorimage forms.
In [Urton] the two ways of tying a figureeight knot in a cord are described as an Sknot (a. and b., front and back) or a Zknot (c. and d.), in analogy with the two directions in which yarn can be spun.
The figureeight knot is different from all the overhand knots in that its two mirrorimage forms are topologically equivalent. The deformation can be achieved by loosening the knots and flipping selected loops over or under the rest of the knot.
The deformation from the Sknot figureeight (form b.) to the Zknot (form c.). In step 1, the red loop is flipped over the rest of the knot; in step 2, the blue loop is flipped under.
Long knots and friar's knots

The 3turn friar's knot (two views), compared with the quipu long knot representing the number 3. These knots (both tied lefthanded here) are topologically the same.


Detail of a fresco (painted 12971300 and sometimes attributed to Giotto) in the Upper Church of San Francesco in Assisi; the saint is shown wearing the habit and cincture of his followers. In particular the cincture has distinctive knots in the hanging end.

Garcilaso de la Vega's Commentarios Reales de los Incas (Lisbon, 1609) describe quipus in some detail. In particular (as quoted in [Locke, p. 40]) "... the knots for each number were made together in one company, like the knots represented in the girdle of the ever blessed Patriarch, St. Francis." Franciscan friars still wear such a belt; traditionally the knots are three, representing vows of poverty, chastity and obedience. In fact, these knots are topologically identical to long knots in quipu, but they are tied so as to be symmetric and consequently do not look the same.
The rule for tying one of the long knots is given in [Locke]: "It is formed by tying the overhand knot and passing the end through the loop of the knot as many times as there are units to be denoted. One end in then drawn taut, thus coiling the other about the strand the required number of times." The rule for tying one of the friar's knots is given in great detail here and is nicely explained by Friar Stephen Agosto of the Orthodox Anglican Society of St. Francis, in this video.

Friar Agosto's method for tying a Franciscan cincture knot: stretch out a length of cord; wind backwards the number of turns wanted; lead the working end of the cord through the tube thus created. As the knot is pulled tight, it must be gently massaged to keep the symmetrical shape.


Friar's knots are sometimes found in quipus: [Locke] gives an example, apparently typical of "the Neighborhood of Cajamarquilla," of a friar's knot with three turns. Locke does not approve of this method: "... the peculiar wrapped condition shown in Fig. 6 [here reproduced on the left]... together with the softness of the cord, produces a knot in which it is hardly possible to note the number of turns." Locke's taut knot is the same as the threeturn friar's knot shown at the start of this section, but tied righthanded.
These knots are also mentioned in [Ashley], under "SingleStrand Stopper or Terminal Knots," as item #517: "A Threefold overhand knot or even a larger one, may be tied, but beyond two turns the knot must be worked into shape, and for that reason it may be considered more decorative than practical." Ashley draws the loose knot as Locke's 4 above (but with 7 crossings), and the compact form exactly as Locke's knot from Cajamarquilla (except with the opposite handedness in both cases).

The equivalence between the quipu knots and the friar's knots with the same number of turns is not completely obvious. To maneuver one to the other in space requires some energetic twisting.
These knot projections show steps in the deformation from the 3ring friar's knot to the torus knot $7_1$ and the quipu knot 3. The middle winding has been colored red for easier reading.

a.  c. The outermost two windings are slid outward, forming two independent loops which are then untwisted. Note that c. shows the same configuration as Locke's Figure 6, except seen from the rear.

c.  f. After additional sliding moves, two more independent loops are formed and untwisted.

f.  h. This part of the deformation is unsymmetric, since there is only one central winding. That part of the cord is slid to the left so as to form one more independent loop, which is twisted away.

h.  j. The resulting configuration is reshaped to resemble the torus knot $7_1$ and then the quipu knot 3.
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The writhe of a knot
The "energetic twisting" required to deform a friar's knot into the corresponding long knot can be quantified mathematically, using the writhe. This is an integer, associated to a knot, that measures its geometrical complexity. The writhe can be defined from a planar projection of the knot; it is not a topological invariant.

The writhe of a knot is calculated from a knot projection diagram by choosing a direction on the string; then (see illustration at left) an integer is asigned to each crossing: $+1$ if the rotation from overcrossing direction to undercrossing direction is to the left (counterclockwise); $1$ if it is to the right. The writhe is the sum of all these integers. It does not depend on the choice of string direction, but changing the handedness (interchanging over and undercrossings throughout the knot) reverses the sign of the writhe.
The writhe is invariant under sliding moves, but changes by $1$ each time an independent loop is formed or eliminated.

Some knots and their writhes. The standard figureeight (without any extra independent loops) has writhe $0$. The overhand knot has writhe $+3$ if it is tied lefthanded, and $3$ if it is tied righthanded.
The friar's knot with three turns (tied righthanded) has writhe $12$, while the topologically equivalent quipu longknot 3 has writhe $7$. In fact during the deformation from one to the other shown in steps a. through j. above, five separate independent loops are untwisted. This accounts exactly for the change in writhe, and explains how the effort required to carry out the deformation between actual knots will vary with the torsional rigidity of the cord.
Comments?
Note: revised, May 5, 2014.
The revision corrects an error in the original account of the topology of the figureeight knot. Thanks to Prof. Ian Agol for bringing the error to our attention.
References

[Ascher] Martha Ascher and Robert Ascher, Code of the Quipu, University of Michigan Press, Ann Arbor, 1981

[Ashley] Clifford W. Ashley, The Ashley Book of Knots. Douibleday, New York, 1944

[Locke] L. Leland Locke, The Ancient Quipu, The American Museum of Natural History, New York, 1923

[Mackey] Carol J. Mackey, Knot Records in Ancient and Modern Peru, Thesis, University of California, Berkeley, Department of Anthropology, 1970. (University Microfilms International, Ann Arbor, 1976

[Urton] Gary Urton, Signs of the Inka Khipu, University of Texas Press, Austin, 2003

[Urton and Brezine] Gary Urton and Carrie J. Brezine, Khipu Accounting in Ancient Peru, Science 309 (2005) 10651067
Tony Phillips
Stony Brook University
Email Tony Phillips