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University of Nevada, Reno Knot Groups and Their Homomorphisms

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University of Nevada, Reno Knot Groups and Their Homomorphisms University of Nevada, Reno Knot Groups and Their Homomorphisms into SU(2) A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics By Rashelle DeBolt Dr. Christopher Herald/Thesis Advisor August, 2019 We recommend that the thesis prepared under our supervision by RASHELLE DEBOLT entitled Knot Groups and their Homomorphisms into SU(2) be accepted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Christopher Herald, Ph.D., Advisor Stanislav Jabuka, Ph.D., Committee Member James Winn, Ph.D., Graduate School Representative David W. Zeh, Ph.D., Dean, Graduate School August, 2019 i Abstract We begin with an introduction to algebraic topology, knot theory, and SU(2) matrices as a subset of the quaternions, then proceed to introduce a technique of finding homomorphisms of knot complement fundamental groups into SU(2) and illustrate it by finding homomorphisms for the fundamental groups of the complement of the trefoil and the complement of the Whitehead link. Finally, the Seifert-van Kampen theorem allows us to find pairs of those homomorphisms, with nonabelian image, which give rise to homomorphisms from the knot group of the Whitehead double and therefore prove that the Whitehead double of the trefoil knot is not the trivial knot. ii Acknowledgments The author is grateful to Dr. Christopher Herald for his advice and encouragement and also to her thesis committee, professors, classmates, and family for all of their support. iii Contents 1 An Introduction to Knots1 2 Free Groups and Finitely Presented Groups5 3 Fundamental Groups 11 4 The Seifert-van Kampen Theorem 20 5 The Wirtinger Presentation 27 6 Meridians and Longitudes of a Knot 33 7 The Whitehead Link 35 8 The Quaternions and SU(2) Matrices 37 9 Homomorphisms of Knot Groups Into SU(2) 45 10 The Whitehead Double of a Knot 51 Bibliography 63 1 1 An Introduction to Knots A knot is an embedding K : S1 ! S3. The complement of a point in S3 is homeomorphic to R3; the homeomorphism S3 n fpg ! R3 can be computed through stereographic projection. Since S3 n fpg is homeomorphic to R3, it is sufficient to picture knots in R3. Definition 1.1 (Definition 1.1 of [2]). Two embeddings f0; f1 : X ! Y are isotopic if there is an embedding F : X ×I ! Y ×I such that F (x; t) = (f(x; t); t) for x 2 X, t 2 I with f(x; 0) = f0(x) and f(x; 1) = f1(x). We call a level- preserving map F an isotopy and frequently use the notation ft(x) for f(x; t) so the map f can be thought of as a path of maps from X to Y . The general notion of Isotopy is not good enough, though, because any two embeddings can be isotopic even if they are different in knottedness. For example, the diagram of the trefoil in Figure1 can be contracted continuously so that the section with all of the crossings is contracted to a point, making it isotopic to the unknot. Figure 1: Definition 1.2 (Definition 1.2 of [2]). Two embeddings f0; f1 : X ! Y are ambient isotopic if there is an isotopy H : Y × I ! Y × I such that H(y; t) = (ht(y); t) with f1 = h1 ◦ f0 and h0 = idY . The difference between the two definitions is that an isotopy moves the set 2 of f0(X) continuously to f1(X) in Y ignoring the neighboring points of Y while an ambient isotopy requires Y to move continuously along with ft(X). In this thesis, we want to restrict our definition of knots to only include to tame knots. Definition 1.3 (Definition 1.3 of [2]). A knot K is called tame if it is ambient isotopic to a simple closed polygon in S3. A knot is wild if it is not tame. Definition 1.4. If two knots in S3 are ambient isotopic, we say they are equiv- alent. The geometric description of a knot in S3 is complicated, but we can use the orthogonal projection of a knot onto a plane in R3, which we will represent in R2 in the following way. Definition 1.5. A knot projection is the image of K : S1 ! S3 n fpg = R3, fpg sufficiently far from K(S1), projected onto a linear subspace E ⊂ R3. The notation K is used to denote the projection of K(S1). Definition 1.6. A knot projection is regular if the map is not injective at only finitely many points and for each of those points, there are no more than two elements in the domain. We call the points at which the regular projection is not injective double points. We also want to insist that at each double point, the arcs actually cross; to do this, we won't allow corners of any ambient isotopic polygonal knots to intersect in the projection. An example of the image of a regular projection of the trefoil is presented in Figure2. 3 Figure 2: In order to represent the image of a knot with a projection in a way that helps us reconstruct the knot in R3 or S3, we need to keep track of the under- and over-crossings at each double point. To do this, we will use the knot diagram: Definition 1.7. A knot diagram is an image of a regular projection which preserves over-crossings and under-crossings in the canonical way so that it is clear how to reconstruct the knot from the two dimensional image. The knot diagrams of a few common knots are shown in Figure3. For Figure 3: more images of common knot diagrams, see appendix D in [2]. Proposition 1.8. For any knot in R3 = S3 n fpg, the set of vectors in S2 for which the orthogonal projection yields a regular projection is open and dense in S2. 4 Proof. The proof of this theorem is outside the scope of this thesis, so we will sketch it out. Sard's theorem states that the set of critical values of a smooth function from one Euclidean space or manifold to another is a null set, which in the case of the set of regular projections, means the set of planes which do not yield regular projections have Lebesgue measure 0. Then, using differential topology, we can show that the set of regular projections is open and dense in S2. Reidemeister was the first to find a finite collection of moves which generate the equivalence relation of ambient isotopy. Theorem 1.9 (Reidemeister Moves). Two knots K0 and K1 are equivalent if and only if they possess diagrams which are related by finitely many Reidemeister moves, Ωi, i = 1; 2; 3. These moves are seen in Figure4 Figure 4: Reidemeister Moves Figure5 walks through an example that shows the figure-eight knot is ambient isotopic to its mirror image. (Note that this is not true in general.) 5 Figure 5: 2 Free Groups and Finitely Presented Groups Like many questions in topology, determining whether two knots are equivalent is difficult without using other mathematical techniques. In particular, we need to introduce some algebraic tools. Free groups are an important ingredient in algebraic topology; however, before we define a free group, we need to talk about algebraic words. Definition 2.1. Let G be a group and let fGαgα2J be a family of subgroups of G. We say these groups generate G if every element g 2 G can be written as a finite product of elements of the groups Gα.A word, w, is a finite sequence of elements from the groups Gα. Furthermore, for any 1 ≤ i ≤ n and αi 2 J, if gi 2 Gαi , the ordered list w = (g1; g2; :::; gn) is a word of length n. We include the empty list, w = ( ), which we call the empty word. 6 Definition 2.2. If fGαgα2J is a family of subgroups which generates G, there is a finite word (g1; :::; gn) of elements of the groups Gα such that g = g1 ··· gn; we say the word (g1; :::; gn) represents the element g 2 G. To combine two words we use the operation of concatenation. So, if 0 0 0 0 w = (g1; g2; :::; gn) and w = (g1; g2; :::; gm) are two words of lengths n and m 0 0 0 0 0 0 0 respectively, then ww = (g1; g2; :::; gn)(g1; g2; :::; gm) = (g1; g2; :::; gn; g1; g2; :::; gm) is also a word of length n + m. We now introduce two operations which will be called reductions of words. If gi and gi+1 are two consecutive entries of a word w and both belong to the same subgroup, then we can combine them to make a word of length n− 1. For example, if w = (g1; g2; :::; gi; gi+1; :::; gn) and gi; gi+1 2 Gαi , we can define a shorter word w0 = (g ; g ; :::; g g ; :::; g ). Moreover, if any g = e is an identity, we can 1 2 i i+1 n i Gαi delete it from the sequence to obtain an n − 1 length word. This can be seen if we define a word w = (g ; g ; :::; e ; :::; g ). Then the identity in the i-th place can 1 2 Gαi n 0 be deleted to obtain the shorter word, w = (g1; g2; :::gi−1; gi+1; :::; gn). Definition 2.3. A reduced word is a word in which no reductions can be made so it is of minimal length. Also, given any word, there is a unique reduced word it can be reduced to.
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