![]() ![]() That is, we pick a singularity of, look at the two ways of splitting it apart, and look at the difference between evaluated on each of the simpler singular knots. Now that we can define for knots with one singularity, we can extend it to knots with any number of singularities inductively. Using the orientation we can make this choice precise, but it requires more knot theory than I am willing to introduce. There is the problem that we did not distinguish between the two ways of resolving the singularity, and so the invariant would seem to be only defined up to sign. Given a singular knot with only one singularity, there are two non-singular knots nearby, corresponding to the two ways of moving the two intersecting pieces of loop apart. Lets say I have a knot invariant for regular knots. Fortunately, there is a canonical way to do this. Now we could try to study this broader class of knots by coming up with new and interesting knot invariants, but a far lazier solution is to try to turn an older knot invariant into an invariant of singular knots. To be more precise, we are allowed only those isotopies between knots which are restrictions of isotopies of the ambient space (hence the name). The new notion of ambient isotopy is basically the same, except that we can’t break a self-intersection two pieces that intersect stay intersected. So a singular knot is like a regular knot, except where the loop can intersect itself in a simple manner. An (oriented) singular knot is a smooth map from into, with only transverse self-intersections (and finitely many of them). The first thing we will do is broaden the concept of a knot. The main tool in doing this is to discover knot invariants, numbers which can be assigned to each knot in a computable way, such that two knots which are the same get assigned the same number. The main problem in the study of knots is to figure out when two knots are definitely the same, or when they are definitely different. Just to review: an (oriented) knot is a smooth inclusion of the loop into, modulo ambient isotopy, which says that we can move the knot around, but we can’t let it pass through itself (ie, it must remain an inclusion throughout the isotopy). If anyone reading is more familiar with some of the places these come up, please send me a reference. They also come up in a number of different areas (solving Feynman diagrams, the representation theory of lie algebras) that I know very little about. Today, though, I’d like to talk about ‘chord diagrams’, a type of object subtly related to that of knots, and whose study can yield some interesting new knot invariants. A good example of this are Tait’s conjectures, three basic conjectures from the 19th century that resisted proof until the discovery of the Jones invariant using techniques from analysis and representation theory (and secretly physics). ![]() I’ve always had a soft spot for knot theory, since its like a poor man’s number theory a source of simple problems which require techniques from advanced math to solve. I’ll try to get back into the swing of things by talking about some stuff I really enjoy, but is far from my research: knots. It also doesn’t help that the more I bend my mind towards research, the less time I spend thinking about things that would actually make good posts (I have started and abandoned several posts on uninteresting research-type things in the last couple months). So, when time and energy get tight, its one of the first things to go. The real problem is that this blog can’t be a higher priority for me than, you know, important stuff like learning, teaching and research. Well, its been quite awhile since the last post.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |