Transcript Maths Presentation

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.
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.
1. Take out (or put in) a
simple twist in the knot:
2. Add or remove two
crossings (lay one strand
over another):
3. Slide a strand from one
side of a crossing to the
other:
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.
Unknot
Trefoil
Figure Eight
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
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
n
Number of prime knots
with n crossings
3 4 5 6 7
1 1 2 3
8
9
10
11
12
7 21 49 165 553 2176
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
(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
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.
Here is a collection of torus knots arranged according to crossing
number