Knot Theory

By Nethania Okyere, Rachel Rozansky, Ashleigh Taylor, and Sylvia Towey

Knot Theory

The knot theory are two mathematical branches of topology. Its simply a loop in 3 dimensional space( doesn’t intersect itself). Knots can be described in various ways. Given a method of description, however, there may be more than one description that represents the same knot. For example, a common method of describing a knot is using a knot diagram. Any given knot can be drawn in many different ways using a knot diagram.

1knots knot theory

Knots math practice of a twisted unknot:

First polynomial:

{1twisted unknot 3} = a{o  } + b{1twisted unknot 2}

= ac{1twisted unknot 2} +b{1twisted unknot 2}

= (ac +b){1twisted unknot 2}

= (ac +b)(a{o o} +b{o})

= (ac +b)(ac{o} +b{o})

= (ac +b)(ac +b)

= (a(-a^2 -a^(-2)) +a^(-1))^2

= (-a^3)^2

{1twisted unknot 3} = a^6

Kaufman polynomial:

{1twisted unknot 3} = (-a)^((-3)(2))(a^6)

{1twisted unknot 3} = 1

Knot Theory

What is a knot? It is an embedding of a circle in 3-dimensional Euclidean space. To make a knot, you must first get string. That is it. With string you can make any type of knot. You can also combine, twist, and analyze them any way possible. But not only can you use them in geometry or topology, but you can use them in chemistry. DNA is actually one unknot that twists and combines to make different models. Another thing in chemistry is that molecular structures can be in the form of knots. For example, one loop of carbon bonds makes an unknot. Other structures might twist and loop and go down and under in many ways. But the one most important rule is no cutting and fusing to make another knot (unless it’s a composite knot). I think it is amazing that knots can play a huge role in many subjects like geometry, topology, and chemistry.

Link: a link is several knots that don’t intersect each other

1Hopf link

Knot Addition

Not only can knots be analyzed but you can do math with them. You can add knots together, and you can factor knots, and there are even prime knots!

Visually, adding knots works like addition, you cut the strings and attach them to each other making a composite knot.

1Adding knots

Mathematically, you add knots by multiplying their polynomials to give the polynomial of the new knot. Because the unknot has a polynomial of one it can be added to any knot without changing it.

Knots can be broken down into their prime factors. Most knots, like numbers, are not prime and these are called composite knots because they can be broken down into prime knots but there are some knots that cannot be broken into smaller knots that are prime. A few examples of prime knots are both the trefoils and the hopf link.

Seifert Surfaces

These are seifert surfaces, a 3d representation of the surface of a knot.Personally I think there should be a line of furniture modeled after these as the beautiful diagrams would make some lovely tables and chairs.

1Seifert surface knot

Braid Theory

The Braid theory is basically the same as the knot theory but has a richer algebraic structure. Personally I found braids easier, but it depends on the person. In knots, you can change it to be a knot that it is equal to. You can do the same for braids. With braids, you have to travel from left to right, you can’t “double-back” as they say.  The braids theory is an abstract geometric theory studying the  everyday braid concept, and some generalizations. The idea is that  braids can be organized as groups, in which the group operation is  using a set of strings to do the first braid and then follow it with a  second braid on the twisted strings. The Braid Theory is very interesting and I had a great time learning about it!

1Braid theory picture

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