# Quandles are associated with knot theory

• stanford1463
In summary, the conversation discusses various concepts related to knot theory, including quandles, shadow colorings, knot invariants, and racks. The participants also mention the difficulty of understanding research papers and suggest resources for learning about algebraic topology, including a book by Hatcher. They also discuss the definition of an unknot and recommend learning about point-set topology and abstract algebra for a better understanding of the subject.
stanford1463
So I know that quandles are associated with knot theory, etc. but what are "shadow colorings" ?

on a broader context, can someone please give me a simple definition of a "knot invariant" and a "quandle" ?

and what does this have to do with "racks" ?
sorry, all this esoteric language is really confusing.
it would be awesome if someone could give me a dumbed down version of knot theory, as an introduction to it. I am kinda interested in this subject, just i need to understand the basic meanings/definitions first..thanks!

i tried reading a bunch of research papers..though I failed epicly. too much vocab. and concepts that I did not understand. for example, "isotopy" and "cocycle" and "quandle coloring"

bump pl0x ?

i don't know if you can learn how to read research papers in a field by just learning all the definitions. For instance, for cocycles to have very much meaning for you, you will want to learn at least a little algebraic topology.

I don't know what a knot really is, but i know that at least some are boundaries of open sets in euclidean space. For instance, the un-knot is the boundary of something which lies entirely in the real plane. But, all the other knots are not this way. Thats at least a starting point.

For isotropy, I imagine this comes from group theory and group actions.

So I was looking at wikipedia. It seems that the unknot is simply a circle in R2? idk.
what are some good websites/sources for learning a bit about algebraic topology? thanks!

bump .

stanford1463 said:
what are some good websites/sources for learning a bit about algebraic topology? thanks!

Hi stanford1463!

Try http://freescience.info/Mathematics.php?id=373

thanks! but is there anyway to bypass all this initial info? I just want to jump straight to learning about quandles and shadow colorings of knots :]

stanford1463 said:
So I was looking at wikipedia. It seems that the unknot is simply a circle in R2? idk.
what are some good websites/sources for learning a bit about algebraic topology? thanks!

exactly, the circle is the boundary of an open disk in R^2.

well, the best source, I would say, is Hatcher's Algebraic Topology: http://www.math.cornell.edu/~hatcher/AT/ATpage.html

But, be warned, in some ways chapter 0 is the hardest in the book. I would definitely start with chapter 1. I was looking at my copy yesterday to refresh on algebraic topology and I noticed that in this chapter he has an example involving knots.

I don't know if any point-set topology would be prereq. for this book. i mean it probably depends on you. i think you should probably know some basic facts about point-set topology first though, but not necessarily at the level of a formal course. The more you know though about point-set topology, abstract algebra, manifolds et cetera, the more you will get out of the book. But, that's not to say you can't get anything out of it right now.

If you wanted to buy a copy, it is very cheap. something like 30 dollars. that's probably the best textbook deal possible.

## What is a quandle?

A quandle is a mathematical structure that is used to study knot theory. It is a set equipped with a binary operation that satisfies three axioms: self-distributivity, self-invertibility, and the unipotent property.

## How are quandles related to knot theory?

Quandles are closely related to knot theory because they provide a way to encode the information of a knot or link diagram in a more algebraic manner. This allows for the use of algebraic techniques to study and classify knots and links.

## What is the significance of the self-distributivity axiom in quandles?

The self-distributivity axiom in quandles is significant because it ensures that the binary operation behaves like a distributive property. This property allows for the manipulation and simplification of quandle equations, making it easier to classify knots and links.

## Are there any real-world applications of quandles?

While quandles were initially developed for their applications in knot theory, they have also found use in other areas of mathematics, including group theory, topology, and algebraic geometry. They have also been applied in computer science and physics.

## How can one construct a quandle from a knot or link diagram?

There are a few different ways to construct a quandle from a knot or link diagram, including the dihedral quandle, the fundamental quandle, and the rack quandle. Each construction method has its own advantages and is used for different purposes in knot theory.

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