![]() ![]() Now that we have Reidemeister's theorem, we can at last construct some invariants and use them to prove that certain knots and links are inequivalent. ![]() A knot invariant is any function i of knots which depends only on their equivalence classes. We have not yet speci ed what kind of values an invariant should take. Welcome to Danielle Roberts and Justin Duffs Wedding Website View photos, directions, registry details and more at The Knot. The most common invariants are integer-valued, but they might havevalues in the rationals Q, a polynomial ring Z, a Laurent polynomial ring (negative powers of x allowed) Z, or even be functions which assign to any knot a group (thought of up to isomorphism). Welcome to Amy Roberts and Justin Stockfords Wedding Website View photos, directions, registry details and more at The Knot. Helpful note: If your search is not generating a list of names, try removing the month and year. Enter the couple's name and wedding date and see your results. The function of an invariant istodistinguish (i.e. If a couple has linked their Wedding Registry (or multiple registries), you can find them by using The Knot's Couple Search Tool. The de nition says that if K = K 0 then i(K) =i(K 0 ). Therefore if i(K) 6= i(K 0 ) then K K 0 cannot be equivalent they have been distinguished by i. Warning: the de nition does not work in reverse: if two knots have equal invariants then they are not necessarily equivalent. topics: Knots and isotopy of knots, Reidemeister theorem, Linking number, p-colourings of knots. As a trivial example, the function i which takes the value 0 on all knots is a valid invariant but which is totally useless! Better examples will be given below. 3-coloring and other elementary invariants of knots, by Jozef Przytycki. Introduction to knots and a survey of knot colorings. The Trieste look at knot theory, by Jozef Przytycki. Introduction to Hopf algebras, by Ken Brown. Link invariants, oriented link invariants, and so on (for all the di erent types of knotty things we might consider) are de ned and used similarly. Close to what we'll cover in the first half of the course. The crossing number c(K) is the minimal number of crossings occurring in any diagram of the knot K. I think the standard recommendations are Adams The Knot Book and Rolfsens Knots and Links, but latter is probably the wrong choice if you dont know some algebraic topology (fundamental group + homology).But there are other good ones. ![]() Page through a few books until you find one you like. This is an invariant by de nition, but at this stage the only crossing number we can actually compute is that of the unknot, namely zero! Example 3.1.7. Your best bet is to look in your institutions library. The number of components (L) of a link L is an invariant (since wiggling via -moves does not change it, it does depend only on the equivalence class of link). De ne the stick number of a knot to be the minimal number of arc segments with which it can be built. Show that the only knots with 4 or 5 arcs are unknots, and show thus that the trefoil has stick number 6. De ne the human number (!) of a knot to be the minimal number of people it takes (holding hands in a chain) to make the knot - what is it for the trefoil and gure-eight? 3.2. One of the simplest invariants that can actually be computed easily is the linking number of an oriented link. It is computed by using a diagram of the link, so we then have to use Reidemeister's theorem to prove that it is independent of this choice of diagram, and consequently really does depend only on the original link. How to get rid of nested circles by changing the knot diagram.Then the total linking number Lk(D) is obtained by taking half the sum, over all crossings, of contributions from each given by +1 ,1 if the two arcs involved in the crossing belong to di erent components of the link, and 0 if they belong to the same one. Different Seifert surfaces from the same knot diagram. Compute the Seifert matrix of the following knot bounding this surface. Seifert's algorithm illustrated on the knot 4_1. The classification of surfaces and surfaces with boundary. Read Sections 2.1, 2.2 and 2.3 of Knots Knotes' by Justin Roberts. An example that shows that the number of loops does not necessarily decrease if you increase the number of minuses in a state. p-colourability is not a complete knot invariant: counterexample An example of a wild knot, which we will exclude from now on. A knot whose unknotting number is realised by first increasing the number of crossings. A convenient set ofsuch rules is called theReidemeister moves, or theR-moves, which act on some small portion of the knot,leaving the rest unchanged. The Alexander polynomial Material from the lecture course The unknot (b) The left trefoil knot (c) The right trefoil knot Figure 1: Knots Since our knots are two-dimensional, we must have rules for manipulating them. Exercise classes: Click here for details regarding the exercise classes and exercise sheets. ![]()
0 Comments
Leave a Reply. |
Details
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |