mathematical knots- laughridge

7/27/2019 Mathematical Knots- Laughridge

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Mathematical Knots

Patterns Unit


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History of Knot Theory Knot Theory: Although knots have been around for thousands of

years, they have only been a particular fascination formathematicians for a little over a century. It is a branch ofmathematics call topology.

In the late 1800's, most scientists believed that the universe was

filled with ether, and all matter was thought to be entangled in it.Lord Kelvin proposed that each element should have uniquesignature based on how the element knotted up the ethersurrounding it. This led many scientists to theorize that they couldunderstand the elements by simply studying knots, somathematicians the world over began to construct tables of knotsand their pictures.

However, soon enough the atomic revolution dismissed the theoryof ether, and mathematicians were left alone in pursuit of KnotTheory for almost a century.


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What is a Mathematical Knot?

A "mathematical" knot is just slightly differentfrom the knots we see and use every day.

First, take a piece of string or rope. Tie a knot in

it. Now, glue or tape the ends together. Youhave created a mathematical knot.

The last step, joining the ends of the rope, iswhat differentiates mathematical knots from

regular knots. The kinds of knotsmathematicians work with are always formed ona closed loop (no loose ends).


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Definition of a Knot

A knot is a simple closed curve in 3-dimensional space.

What does that mean? Well, a loop isconsidered a knot in mathematical knot

theory (it is a simple closed curve in 3-

dimensional space). In fact this knot has aspecial name: The unknot. It is the

simplest of all knots.


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Basic Vocabulary

The Unknot: is a closed loop of ropewithout a knot in it. The unknot is also

called the trivial knot.


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Basic Vocabulary cont.

Prime Knot: in knot theory, this is a knot that is not thesum of simpler knots.

Examples of prime knots include:

Ambient Isotopy- deformation of a knot.

Invariant Isotopy- the property that states if a knot or

link remains constant regardless of ambient isotopy.


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Vocabulary

Composite knot- can be made by joining 2 or more

non-trivial knots.

Trefoil knot- the simplest form of a non-trivial knot. Thetrefoil knot can be made by joining together loose ends

of a common overhand knot and looping them together.

Good website for information, background, and equationsthat apply to knots. http://library.thinkquest.org/12295/
http://library.thinkquest.org/12295/http://library.thinkquest.org/12295/http://library.thinkquest.org/12295/


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Reidemeister Moves

Finally, German mathematician Kurt

Reidemeister (1893-1971) proved that all

the different transformations on knots

could be described in terms of threesimple moves.

This illustrates a very important feature of mathematics: we

reduce a complicated process to a sequence of simple

steps

You can remove, insert or

change some ofthe crossings

according to Reidemeister

Moves.


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More Basic Vocabulary One problem with knots is knowing whether or not you have one or

just the unknot in disguise. Tricolorability allows you to test knots. Tricolorability: the ability of a knot to be colored with three colors

subject to certain rules.

Rules: (1) At least two colors must be used; and (2) At eachcrossing, the three incident strands are either all the same color or

all different colors.

We will say that a knot or link is tricolorableif each of its strands can be colored one ofthree different colors, so that at each crossing,

either three different colors come togetheror all the same color comes together. Atleast two colors must be used.


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Examples of Tricolorability and Not

Tricolorability

This figure eight knot is NOT tricolorable.

The granny knot is tricolorable. In

this coloring the three strands at

every crossing have three different

colors.


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Tricolorability cont.

Look at the Knots handout.

Find one knot that shows tricolorablity andone that does not. (The knots you find

must be different from the knots on the

previous slide) :o)


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The Central Problem of Knot

Theory The central problem of Knot Theory is determining whether two

knots can be rearranged (without cutting) to be exactly alike.

A special case of this problem is one of the fundamental questionsof Knot Theory: Given a knot, is it the unknot?Now, for a simple

loop, that's an easy question. Is it possible to transform this knotso that it looks like the unknot? Tie a trefoil knot yourself and seeif you can untangle it to form a simple circular loop.

When we actually start trying to untangle and rearrange knots tolook like one another, we begin what can seem like a very

complicated process. Mathematicians were perplexed at theseemingly unending number of ways a knot could be shaped andturned. What was needed was a simple set of rules for working withknots.


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Knot Challenges

One very interesting knot design is known as the

Borromean Rings. This design appears on the

Coat of Arms of the Italian Renaissance family

Borromeo. The knot design consists of threerings. No two are connected, yet the three

together cannot be separated. They are linked

together in such a way that if any one ring is

removed from the set of three, the two remainingrings are no longer connected. Got that? Good.

Sketch the Borromean Rings.


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The Borromean Rings


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