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Knots have been studied extensively by mathematicians for the last hundred years. One of the most peculiar things which emerges as you study knots is how a category of objects as simple as a knot could be so rich in profound mathematical connections Knot Theory is the mathematical study of knots. A mathematical knot has no loose or dangling ends; the ends are joined to form a single twisted loop. Knots

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The Reidemeister Moves Reidemeister moves change the projection of the knot. This in turn, changes the relation between crossings, but does not change the knot. 3. Slide a strand from one side of a crossing to the other: 2. Add or remove two crossings (lay one strand over another): 1.Take out (or put in) a simple twist in the knot:

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Famous Knots TrefoilFigure EightUnknot In order to talk mathematically about knots, I have to show them in some kind of way, to have a method of describing them. I did this for the simplest knots by using a piece of string or rope, which nicely shows the 3-dimensional nature of the object. Here are some of the most famous knots, all known to be inequivalent. In other words, none of these three can be rearranged to look like the others. However, proving this fact is difficult. This is where the mathematics comes in.

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Crossings – What are they? 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. A zero knot has 0 crossings A trefoil knot has 3 crossings. On the left shows a picture with 9 crossings

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Prime Knots The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,.... Any number can be written as a product of a set of prime numbers. Here is an example: 60 = 2 x 2 x 3 x 5 = 2 x 5 x 3 x 2. The number 60 determines the list 2, 2, 3, 5 of primes, but not the order in which they are used. The same is true for knots. A prime knot is one that is not the sum of simpler knots. To work out the number of knots with a number of crossings, a table is given below where it compares the number of prime knots against crossing number n3456789101112 Number of prime knots with n crossings 1123721491655532176

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Torus Knots – What are they? Torus is the mathematical name for an inner tube or doughnut. It is a special kind of knot which lies on the surface of an unknotted torus. Each torus knot is specified by a pair of coprime integers p and q. The (p,q)-torus knot winds q times around a circle inside the torus, which goes all the way around the torus, and p times around a line through the hole in the torus, which passes once through the hole, On the left/right shows a picture called (15,4) torus knot because it is wrapped 15 times one way and 4 times the other

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(p,q) torus knots The (p,q)-torus knot can be given by the parameterization This lies on the surface of the torus given by (r − 2)2 + z2 = 1

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Arithmetic of knots From this we can make a general rule about the addition of knots: K + L = L + K. This is called commutativity. Below shows how you add 2 knots together

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Here is a collection of torus knots arranged according to crossing number

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