MME 529 June 2017 Canonical Forms Some questions to be explored by high school investigators William J. Martin, WPI Here are some exercises based on various ideas of canonical form in mathematics. Perhaps a few can be used with students. I think we subconsciously use canonical forms in many parts of mathematics, mainly as a sort of language tool. In order to communicate with one another, we must agree to write things in the same way, as we agree on spelling conventions. Rational numbers are written with relatively prime numerator and denominator, with the denominator always positive; algebraic numbers never have square roots in their denominators; integers in modular arithmetic are always reduced to their smallest non-negative remainder; etc. But the importance of canonical forms can be lost if we simply view them as notational conventions. The more important point is that canonical forms give us algorithms to determine if two mathematical objects are the same. If two students have fractions such as 105/35 and 51/( 17), one way to have them determine if the numbers are equal is to ask each student to reduce their fraction: they will both arrive at 3/1 and will immediately see that they are equal. We also do this if we are comparing f(x) = x 3 13x+12 against g(x) = (x 3)(x 1)(x+4) or h(x) = (x + 4)(x 2 + 2x 3). Expanding the three polynomials and writing them each in the canonical form a 3 x 3 + a 2 x 2 + a 1 x + a 0, we may quickly check which two are equal. Analogously, suppose you and your friend are trying to locate one another on a busy summer day at Quincy Market in Boston. Supppose you are just a few hundred yards away from one another and are in touch via cell phone. She might say she is next to the T-shirt salesman and you might say that she can easily find you next to the busker. One way to meet up is to agree that both of you will go to the Sam Adams statue just west of Faneuil Hall. There is exactly one Sam Adams statue and you can both find it easily. Isn t that sort of what reducing fractions is like? Rather than either one of you finding the other, you agree to move to an agreed-upon location. With this in mind, here is a worksheet about canonical forms in arithmetic, linear algebra, number theory (with a brief excursion into an easy-to-explain cryptosystem), graph theory and knot theory. 1 Not so hard Write these numbers in standard form: 1. a = 18 12 2. b = 3 8 2 3 1
7 3. c = 3 2 In each case, two of the three numbers are equal; reduce to some canonical form to determine which two. 4. a = 18 12, A = 24 9, α = 45 30 5. b = 3 8 2 3, B = 4 27 6 8, β = 6 12 3/2 18 7 6. c = 3 C = 11 + 72 2 3 + σ = 3 2 2 In arithmetic modulo 12, write the result of each computation in canonical form (i.e., using only one of 0, 1,..., 11): 7. 8 + 8 8. 8 8 9. ( 7 3) + ( 5 5) The next few problems involve planes and lines in three-space. 10. Show that the parametric system {(x, y, z) x = 2s 4t, y = s + 4t, z = s t for some s, t R} describes the same plane as {(x, y, z) x + 2y + 4z = 0} 11. The equations 7x 2y + 13z = 0 and 14x + 4y 26z = 0 describe the same plane since the two normal vectors are parallel: [ 7, 2, 13] = 1 [14, 4, 26]. 2 So which among these equations describe the same plane? 12. How about these? 4x 6y 10z = 0, 2x + 3z = 5y, 5z = 2x 3y. 2 3 x + 5y 5 3 z = 4, 2 3 x + 5y = 12 + 5 3 z, 3y z 2 = 2 (x + 1). 5 13. Given are three systems of two equations, each system aligned vertically. Use the reduced row echelon form (rref) button on a calculator to determine which two of these systems describe the same line through the origin in R 3 : x + y + z = 0, 2x y + 8z = 0, x + 3z = 0 2x + 4y 2z = 0, 3x + 2y 8z = 0, y 2z = 0 2
14. Suppose we are now given descriptions of three lines in R 3 l 1 : 3x + 2y z = 5, x y + 5z = 18 ; l 2 : 2x + 2y 11z = 45, 7x + 4y = 20 ; l 3 : 3x + 2y z = 5, x + 2z = 10. (Sorry now the pairs of equations describing each line are aligned horizontally.) Two of these three systems describe exactly the same line. Which two? 2 A Simple Cryptosystem Alice and Bob agree on a secret modulus m and send only messages in the number system R = {0, 1, 2, 3,..., m 1} Their algorithms are as follows. Encrypt: To encrypt message x R, generate a random integer k and send ciphertext c = x + km Decrypt: To decrypt ciphertext c, reduce modulo the secret m to find the unique x in R with x = c (i.e., x c mod m ) Now suppose that Alice encrypts x = 5 as c = 2007 and encrypts x = 8 as c = 190. What can you say about her secret integer m? If you also know that Alice could encrypt x = 3 as c = 725 and x = 2 as c = 2290, can you decipher this message?: 129, 223, 109, 85, 249, 280, 347, 32 How would you attack Alice s system if you knew none of her messages, but only a sequence of millions of ciphertexts? 3 Hard 1. Almost all equations (in variables x and y) of the form Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 are conic sections the shape of the solution set is obtained by intersecting some plane with the cone x 2 + y 2 = z 2. But not all of them have this form.for what values of A, B, C, D, E, F is the solution set not of such shape? 2. The six diagrams in Figure 2 represent only three distinct graphs up to isomorphism. Which ones are equivalent? 3. Use Read s canonization method to reduce each of the 12 trees in Figure 3 to canonical form. Are they all different? 3
4. Carefully write down the sequence of Reidemeister moves (see below) that reduce the first knot diagram in Figure 4 to the second one. (The second one is the right-handed trefoil.) 5. Can you find a short sequence of Reidemeister moves that transform the knot diagram in Figure 5 into the unknot (a simple loop, or O, in three-space)? 6. Research the unknotting problem, starting with the introduction on wikipedia. 4 Read s Canonical Form for Trees Ron Read s naming convention for trees is described in Wikipedia (see https://en.wikipedia.org/wiki/graph canonization). The algorithm gives a name to each tree in such a way that two trees receive the same name if and only if they are isomorphic (i.e., are drawings of the same tree). This name is a very long string of zeros and ones. To understand the algorithm, you need to know a few pieces of terminology. The empty string ε is the unique string of length zero. With the convention that 0 comes before 1 in our alphabet of two letters, we can put all strings in lexicographic order (dictionary order). For strings of length four or less, this order is ε < 0 < 00 < 000 < 0000 < 0001 < 001 < 0010 < 0011 < 01 < 010 < 0100 < 0101 < 011 < 0110 < 0111 < 1 < 10 < 100 < 1000 < 1001 < 101 < 1010 < 1011 < 11 < 110 < 1100 < 1101 < 111 < 1110 < 1111 Next we learn to concatenate strings by just gluing them one after the other. For example, the concatenation of ε, 0011 and 01 (in that order) is 001101 (since the empty string becomes invisible). In a graph, two vertices are adjacent if they are joined by an edge. A leaf in a tree is a vertex which is adjacent to only one other vertex. We begin building our name by labelling every node in the tree with the string 01. Then we keep repeating the following step until only one or two nodes remain: choose any non-leaf vertex; call it v. Delete the leading 0 and trailing 1 from the current label of v. Then build the list of this label and the labels of all leaves adjacent to v. (This is really boring if v is not adjacent to any leaves; so don t choose such a v.) Sort this list in lexicographic order. Now concatenate all labels in the sorted list, add a 0 to the front of this long string and tag a 1 onto the end of it. Make this the new label for node v and delete all leaves adjacent to v. Choosing v appropriately, this reduces the number of vertices by one or more in each step. If we reach a graph with only one vertex, we take the label of this vertex as the name of the original tree. If we end with two adjacent vertices, we put their labels in lexicographic order, concatenate these and take this concatenated string as the name of our tree. Read proved that two trees get the same name if and only if they are isomorphic. (What can you learn about a tree from seeing only its name?) 4
5 Knots and Reidemeister Moves A knot is a drawing of a circle in 3-space; i.e., a continuous one-to-one function from the unit circle to R 3. (To be precise, we should rule out wild knots by making our function piecewise linear.) Knots are represented by knot diagrams which are drawings in the plane with each crossing clearly indicating which of two strands is on top and which is underneath. The same knot can be drawn in infinitely many different ways via knot diagrams, but we say that all these diagrams are equivalent. In general, it is extremely hard to determine if two knot diagrams are equivalent, or even to decide if a knot diagram is a drawing of the unknot O. In 1927, Kurt Reidemeister proved that if two diagrams are diagrams of the same knot, then there are three simple diagram manipulations that can be used to transform one into the other. (If you are familiar with linear algebra, liken these to the three basic row operations on a matrix.) These Reidemeister moves are Type I: creating or removing a loop Type II: sliding one strand across another or sliding them apart Type III: sliding a strand across a crossing of two other strands and are depicted in Figure 1. These are explained further on websites such as wikipedia. Reidemeister s amazing theorem is that two diagrams are diagrams of the same knot if and only if! one can be transformed into the other by a finite sequence of Reidemeister moves. But unfortunately, no one has ever found a canonical form for knot diagrams, so we still do not know how to carry out these Reidemeister moves to efficiently demonstrate such an equivalence. 5
Figure 1: The three Reidemeister moves for transforming knot diagrams. Figure 2: Six graphs on 10 vertices, but really only three different ones. 6
Figure 3: Twelve trees on 7 vertices. Figure 4: Two knot diagrams. 7
Figure 5: Can you untie this knot using Reidemeister moves? 8