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The mathematical theory of knots originated in the 19th century, but knots have been of interest since ancient times. Knots appear in illuminated manuscripts, sculpture, painting and other art forms from all over the world. As early as human beings used any kind of rope, they probably began inventing knots, and sailors and scouts alike can attest to their variety and usefulness. The mathematical theory of knots has made major advances in the past decade. One of the most exciting developments has been the discovery of deep connections between knot theory and the branch of physics that studies the fundamental particles and forces that are the building blocks of the universe. It has also been found that DNA is sometimes knotted, and knots may play a role in molecular biology. A knot is a mathematical object , just like number is, and mathematicians ask many of the same questions about knots as they have asked about numbers. One of these questions is, "Are these two knots equal?"
A mathematical knot has no loose or dangling ends;
the ends are joined to form a single twisted loop.
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Some of the ways that knots are classified
is by the arrangement of their crossings, the characteristics
of their mirror images, and the
braids from which they are formed.
A trefoil knot has 3 crossings. The trefoil knot is so named because of its three loops: tre is related to the word three foil is related to the word leaf Indeed, the trefoil knot bears a resemblance to a 3-leaf clover. The trefoil knot can be formed in two ways. The two versions of the knot differ only in the over/under placement of the strands. |
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| Only the zero knot has a crossing number of zero. | |||
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The zero knot is the knot formed when the
ends of a piece of rope are joined without
any twists or crossings. It even looks like a zero!
But you can twist the zero knot around so that it has crossings in it. Each of the places in a knot where 2 strands touch and one passes over (or under) the other is called a crossing . The number of crossings in a knot is called the crossing number. How useful is the crossing number for classifying knots? Knots have been catalogued in order of increasing complexity. One measure of complexity that is often used is the crossing number, or the number of double points in the simplest planar projection of the knot. There is only one knot with crossing number three (ignoring mirror reflections), the trefoil or cloverleaf knot. The figure-8 knot is the only knot with a crossing number of four. There are two knots with a crossing number of five, three with a crossing number of six, and seven knots with a crossing number of seven. From there on the numbers increase dramatically. There are 12,965 knots with 13 or fewer crossings in a minimal projection and 1,701,935 with 16 or fewer crossings. Following are pictures of the sixteen simplest knots: |
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| Knots such as the square knot are usually excluded from knot tables because they can be constructed of simpler knots. Knots that cannot be split into two or more simpler knots are called prime. | |||
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| When a zero is added to a number, the result is the same number. Similarly, when the zero knot is added to another knot, the result is that same knot, unchanged, except for the addition of some extra rope. If you hold a mirror up to one trefoil knot and look at its reflection, what you see in the mirror is the other trefoil knot! The two trefoil knots are called mirror images of each other. | |||
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| The two trefoil knots are different, but not all knots are different from their mirror image: | |||
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| To be continued... | Back to My Research | ||