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ozarque mentioned knot theory on her LJ. By coincidence, a week or so later I attended a presentation on the subject. It's not my area, but I'd like to try to consolidate my knowledge, and one of the best ways of doing that is, of course, to try to explain it to someone else. Hence, this post. I'm not going to go into technical detail; I'd just like to describe some of the underlying ideas - which sets of tools have been brought to bear, and why.
There are several reasons why mathematicians have become interested in knots. There are significant applications in physics, and also in the geometry of three-dimensional objects; but more than that ("more" in the eyes of any pure mathematician, that is), they are beautiful and intriguingly complex in their own right - that is, they deserve study as ends, not merely as means. Now, one of the first questions one asks is, "When are two of these things 'the same', and how can we tell?" (Okay, that's two questions. Whatever.) I'll talk about the first question in this post, and a little bit about the second in another.
( The same, or not? )
Next time, I'll talk about knot invariants. (If you've read any of my earlier mathematical posts, you knew that invariants would be on the menu, didn't you?)
ozarque mentioned knot theory on her LJ. By coincidence, a week or so later I attended a presentation on the subject. It's not my area, but I'd like to try to consolidate my knowledge, and one of the best ways of doing that is, of course, to try to explain it to someone else. Hence, this post. I'm not going to go into technical detail; I'd just like to describe some of the underlying ideas - which sets of tools have been brought to bear, and why.There are several reasons why mathematicians have become interested in knots. There are significant applications in physics, and also in the geometry of three-dimensional objects; but more than that ("more" in the eyes of any pure mathematician, that is), they are beautiful and intriguingly complex in their own right - that is, they deserve study as ends, not merely as means. Now, one of the first questions one asks is, "When are two of these things 'the same', and how can we tell?" (Okay, that's two questions. Whatever.) I'll talk about the first question in this post, and a little bit about the second in another.
( The same, or not? )
Next time, I'll talk about knot invariants. (If you've read any of my earlier mathematical posts, you knew that invariants would be on the menu, didn't you?)
