Maths In Knots

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Maths In Knots
BY Punam Mistry
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Knots
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.
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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.
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Famous Knots
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.
Trefoil
Figure Eight
Unknot
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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 trefoil knot has 3 crossings.
A zero knot has 0 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
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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
Below shows how you add 2 knots together
From this we can make a general rule about the
addition of knots K L L K.This is called
commutativity.
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Here is a collection of torus knots arranged
according to crossing number