So I know that quandles are associated with knot theory, etc. but what are "shadow colorings" ? on a broader context, can someone plz 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. im 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"
Re: Quandles 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.
Re: Quandles 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!
Re: Quandles 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 :]
Re: Quandles 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, thats 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. thats probably the best textbook deal possible.