C H A P T E R 5

Knotted and Linked Bracelets

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© 2014 by Taylor & Francis Group, LLC

KNOTTED AND LINKED BRACELETS ● 69

In the last chapter, we began by considering what would happen if we wrapped a string around a torus and fastened the ends together to obtain a torus knot.

If we dispense with the torus and explore what happens when we take a length of string, rearrange it any way we like, and fasten the ends together, then we enter the more general realm of knot theory, a fascinating branch of topology. The field originated in the 1800s, inspired by a beautifully imaginative but utterly incorrect theory about the nature of the chemical elements.* The study of knots continued long after scientists came to a proper understanding of the elements, and it blossomed into its own field of inquiry. After all, many mathematicians don’t care whether a subject has obvious practical applica-tions. They are spurred on by the challenge and intrigue of discovery for its own sake, and they understand that a field of mathematics that seems completely useless when it is developed is often unexpectedly useful later on. In recent decades, the story has come full circle with the discovery of exciting new connections between knot theory, genetics, and molecular chemistry.

For a student of knot theory, bead crochet bracelets offer a new avenue of fun exploration because they are, in many ways, the perfect medium for modeling knots. And, of course, they also offer yet another way to adorn your-self with a mathematical model! The primary focus of knot theory is to understand, categorize, and identify different knots. A knot is a strand that is (perhaps) tangled in some way and then connected at the ends. A bead crochet invis-ible join provides the perfect way of creating such a con-nection with a smooth seamlessness that helps capture the essence of a mathematical knot. Unlike a piece of tied string, bead crochet bracelets have no obstacle at the join to interfere with manipulating and exploring the knot in its varied forms. Additionally, the slipperiness of glass or metal beads facilitates easy manipulation between the different forms of a knot. A mathematical knot can be rearranged in any way possible and still be considered the same knot, as long as it is never cut. For example, you may find it surpris-ing that the purple knots and the blue knots in Figure 5.1 are all different incarnations of the same knot, the trefoil (which you may recall was introduced in Chapter 4 as a type of torus knot). Careful inspection also reveals that the purple and blue knots are subtly different versions of the trefoil. In each configuration, whenever the strand of each knot crosses itself, the part of the strand that is on the top in the purple knot is on the bottom in the blue knot, creating

* Colin Adams, The Knot Book, American Mathematical Society, 2004.

a kind of left- and right-handed trait. Color aside, no matter how you manipulate them, you will never be able to make the purple knot look exactly like the blue one! They are fun-damentally different creatures. These kinds of fundamental traits are what knot theorists seek to identify.

One useful way of identifying a knot is to manipulate it into a form with the least possible number of crossings, or places where the strand crosses over or under itself. This number is the minimum crossing number of the knot, and it is the sort of fundamental property that helps topologists distinguish one knot from another. The simplest knot is a plain ring (such as a normal bead crochet bracelet), which has zero crossings and is called the unknot. There is no knot with a minimum crossing number of one or two; you might attempt drawing one to convince yourself of this. The tre-foil, with three crossings, turns out to be the simplest after the unknot, and it is also the only three-crossing knot, although as discussed before, it has left- and right-handed forms. Figure 5.2 shows all the distinct knots (disregarding handedness) up to a minimum crossing number of seven.

Bead crochet knots, depending on length, can be worn as either necklaces or bracelets and, since there are multiple choices for which hole the head or hand might go through, they provide additional fun design choices. Although this does not involve the pattern design techniques we discuss in the other chapters, we couldn’t resist including this addi-tional lovely connection between mathematics and bead crochet, a connection that inspired this set of challenges.

Challenge Can you make a knotted trefoil bracelet and use it to prove that all the bracelets depicted in Figure 5.1 are also trefoils? Can you create a set of bead crochet bracelets or necklaces representing each of the knots up to seven crossings, as shown in Figure 5.2?

You may notice that some of the knots in Figure 5.2 are tied (no pun intended) to the designs from Chapter 4. In fact, three of the knots with seven or fewer crossings (not counting the unknot) are torus knots. Of course, one of them is the trefoil, which we have already identified as a (3,2) torus knot. Can you spot the other two torus knots in Figure 5.2?

Another interesting characteristic of every knot is its braid representation, which can be useful to consider when designing knotted bracelets. In mathematics, a braid is a set of N strands connected to two horizontal bars, one at the top and one at the bottom; the strands may cross over or under one another while moving from the top bar to the bottom one. As illustrated in the braid on the left in Figure 5.3, each strand must always travel downward from

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FIGURE 5.1 Left- and right-handed trefoil knots and some of their many forms. these bracelets are made with size 8 seed beads in circumference 5-around (4-around works well too!) with a simple alternating bead color pattern.

FIGURE 5.2 the knots up to seven crossings. these images were produced using the Knotplot software written by robert Sharein.

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KNOTTED AND LINKED BRACELETS ● 71

the top (so no doubling back upward). A strand may land in a new position on the bottom bar by the time it reaches the bottom. Every knot has at least one braid representation. For example, Figure 5.3 shows a knot called the figure-eight knot (on the right, and also shown bottom left in the chap-ter header photograph) and a braid representation of it (on the left) in a three-strand braid. If we extend the strands beyond the top bar and connect them to the strands at the bottom bar as in the middle illustration of Figure 5.3, we find a closed braid representation of the figure-eight knot. A closed braid representation is simply another configura-tion of the original knot.

A closed braid representation for a knot presents a nice way to wear it, allowing the knotted bracelet to wrap around your wrist in a single direction (always clockwise or always counterclockwise). The number of strands in the braid representation is the number of times the bracelet winds around in that configuration, which helps determine how long to make a bracelet if you wish to wear it this way. For instance, to wear a figure-eight knotted bracelet in the closed braid representation depicted in Figure 5.3, you would make a bracelet a little more than three times the length of a regular bead crochet bracelet; you need a bit of extra length to accommodate the over and under crossings in the braid. Figure 5.4 shows two arrangements of a fig-ure-eight knotted bracelet: the three-strand closed braid representation on the top, and a nonbraid necklace form on the bottom.

We can further generalize the ideas of knot theory by considering what happens when we use more than one strand to make a link. The name is logical, since links are simply knots that are linked together in some way. Knot theorists also seek ways of understanding, identifying,

and categorizing links. One obvious feature of a link is the number of component knots it contains, and another is minimum crossing number—just as for individual knots. Figure 5.5 shows links with up to six crossings. The link on the bottom left is a famous three-component link called the Borromean Rings that has the amusing property that if any one of the three components is removed, the other two components are unlinked. Links offer another fasci-nating approach to jewelry design with bead crochet. The bracelet shown in Figure 5.6 is composed of a trefoil linked with the unknot, forming a blue (3,2) torus knot wrapped around an actual torus, with the brown and green bracelet playing the role of the torus. Depending on the lengths of the different components, linked bracelets can be worn in an endless variety of ways. Conveniently, every link also has a closed braid representation.

FIGuRE 5.3 The figure-eight knot (right), in a braid representa-tion (left), and in a closed braid representation (middle).

FIGuRE 5.4 Two ways of wearing the figure-eight knot from bottom left of the chapter header page. On the top is the three-strand closed braid representation shown in Figure 5.3 (middle), which works well for a bracelet. On the bottom, Teddy is model-ing a nice form for a necklace.

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As another challenge, we suggest investigating the pos-sibilities for creating linked bead crochet bracelets to form interesting jewelry. Readers interested in seeking inspiration for other possible knotted and linked bracelet ideas, or who simply wish to learn more about knot theory, are enthusi-astically urged to consult one of our favorite resources: The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots by Colin C. Adams. Knotted and linked bracelets make a wonderful teaching aid for a class in knot theory. If you happen to be a knot theory student, we can’t imagine a more appropriate class contribution than a set of bead crochet knots and links, which can be used to adorn a wall as easily as a wrist, and which will likely be employed for years to come as treasured classroom manipulatives.

Celtic Knots: Another Design SpringboardOne more interesting knot design motif is the Celtic knot, as explored in the two bracelet designs shown in Figure 5.7. These designs simply paint a picture of a knot

or link using color or blank space to indicate the “string” passing over or under itself. Celtic knots have a rich his-tory in art throughout the ages, and a little research will reveal many beautiful knot designs that can be used as the basis for bead crochet patterns. We hope our readers will be inspired to explore the use of these motifs further.

FIGuRE 5.5 The links up to six crossings. These images were produced using the KnotPlot software written by Robert Sharein.

FIGuRE 5.6 A linked bracelet.

FIGuRE 5.7 Celtic knot designs. The patterns for these bracelets are provided on pp. 163–164.

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