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GENERALIZED P-COLORINGS of KNOTS GENERALIZED P -COLORINGS OF KNOTS Mark Medwid A Thesis Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of MASTER OF ARTS May 2014 Committee: Mihai Staic, Advisor Rieuwert Blok Juan Bes ii ABSTRACT Mihai Staic, Advisor The concept of p-colorings was originally developed by R.H. Fox. Consideration of this knot invariant can range from the simple intuitive definitions to the more sophisticated. In this paper ∼ the notion of abstract colorings by the group Tp = (Zp × Zp) o C3 is developed and explored. The abstract coloring by Tp yields some close connections to the Alexander polynomial, and so Fox’s “free calculus” construction of the Alexander polynomial is closely examined as well. The advantage of this calculation is that it allows the Alexander polynomial to be easily constructed once one knows the knot group of a particular knot. Due to the way C3 acts on (Zp × Zp) in Tp, calculating Tp-colorability amounts to some basic algebraic manipulation. This can also be combined with the linear algebra approach of calculating the exact number of Tp colorings to yield a potentially stronger invariant. iii ACKNOWLEDGEMENTS I would like to thank my fiancee,´ Heather, for her love and care during my times of utmost stress. I would also like to thank my parents, Mark and Barbara, without whom I would be lost. I would also like to thank Rieuwert Blok and Juan Bes for serving on my committee and for all of their helpful comments and suggestions. My colleagues also deserve a special thanks for supporting me on the path to writing this paper. There are too many to thank in the bounds of this page, but I will mention a few: Robert Kelvey, John Haman, Leonardo Pinheiro, Nathan Baxter, Jacob Laubacher, David Walmsley and everyone who attended my defense. Finally, a special thanks to Mihai Staic for helping me with this thesis and encouraging me to write it. It has truly been a worthwhile experience, and his hard work as my advisor made this thesis not only possible, but a piece of writing that I am proud of. iv TABLE OF CONTENTS CHAPTER 1: INTRODUCTION TO KNOTS 1 1.1 KNOTS AND KNOT EQUIVALENCE . .1 1.2 KNOT INVARIANTS . .5 1.3 KNOT GROUPS . .7 CHAPTER 2: COLORINGS OF KNOTS 11 2.1 TRICOLORABILITY . 11 2.2 COLORABILITY: THE LINEAR ALGEBRA APPROACH . 13 2.3 COLORABILITY: THE GROUP-THEORETIC APPROACH . 17 CHAPTER 3: THE ALEXANDER POLYNOMIAL 24 3.1 PRESENTATIONS AND PRESENTATION EQUIVALENCE . 24 3.2 FOX DERIVATIVES AND ELEMENTARY IDEALS . 27 3.3 KNOT POLYNOMIALS . 36 CHAPTER 4: Tp COLORINGS OF KNOTS 41 4.1 MOTIVATION . 41 4.2 THE GROUP Tp ................................... 44 4.3 Tp COLORINGS . 45 4.4 Tp AND THE ALEXANDER POLYNOMIAL . 49 BIBLIOGRAPHY 58 v LIST OF FIGURES 1.1 A diagram of the trefoil knot. .2 1.2 A diagram representing the unknot. .2 1.3 Two diagrams of the unknot. .2 1.4 Diagrams of two different knots. .5 2.1 A tricoloring of the trefoil knot. 11 1 CHAPTER 1: INTRODUCTION TO KNOTS 1.1 KNOTS AND KNOT EQUIVALENCE Humans have often utilized knots for various purposes: stabilizing a load on a length of rope, sealing bags, and many others. The widely used knots are even named – the trefoil knot and granny knot, for instance. In their own right, knots serve as an interesting (and useful!) object worthy of mathematical study. To begin such study, we must formalize the definition of a knot. For the purposes of this thesis, the following definition is used: Definition 1.1. A knot is any embedding K : S1 ! R3. Intuitively, a knot is a closed loop of string in Euclidean space. Though knots are three- dimensional objects and can be visualized, it is sometimes difficult or cumbersome to render com- plicated knots in three dimensions. We often resort to working with drawings which are “good enough” visualizations of knots. Definition 1.2. A knot diagram is a projection of a knot K to the plane. That is, a knot diagram is a two-dimensional representation of a particular knot. In Figure 1.1 below we have a diagram representing one of the trefoil knots. It is a matter of convention that the breaks in the arcs represent an arc going underneath another arc (in other words, it is the underside of a crossing). Some natural questions arise. When can we consider two knots to be the same? How do we distinguish two knots which are different? When is a knot the simplest? These notions shall be formalized shortly, but it makes intuitive sense that two knots are equivalent when they can be 2 Figure 1.1: A diagram of the trefoil knot. stretched, tangled or untangled to match one another without cutting and gluing. The “simplest” knot should be one without any tangling whatsoever. It is referred to as the unknot – a circle without any kinks or crossings. Figure 1.2: A diagram representing the unknot. Our use of diagrams may introduce additional issues. It is possible that one knot may have many different-looking diagrams, such as in Figure 1.3. Is it enough to compare two particular Figure 1.3: Two diagrams of the unknot. diagrams of knots rather than the knots themselves? The answer is contained in the theorem 3 below, which also formalizes the idea of knot equivalence. Theorem 1.3. (Reidemeister’s Theorem) Two knots, K and K0 with diagrams D; D0 are equivalent if and only if their diagrams are related by a finite sequence of intermediate diagrams such that each differs from its predecessor by one of the following four elementary operations. (Each operation is only applied to a specific area of the diagram.) 1. (R0) A planar isotopy. 2. (R1) “Un-kinking” an arc. 3. (R2) Pulling apart two arcs where one is completely above the other. 4. (R3) Pulling an arc over or under an intersection, provided it is completely above or below. 4 These elementary operations are known as the Reidemeister moves. The theorem says that if we can go from one knot diagram to another by a finite sequence of these Reidemeister moves, then both diagrams in fact represent the same knot. The next result is immediate: Proposition 1.4. “Equivalence through a finite sequence of diagrams,” denoted as “∼,” is an equiv- alence relation. Proof. Suppose K is a knot, and D is a diagram of K. Then K ∼ K by the sequence D = D0 = D; a sequence of length one. So ∼ is reflexive. We can see that ∼ is symmetric: sup- 0 0 0 pose K ∼ K by the sequence D = D0;D1;:::;Dn = D . Then we also have that D = 0 Dn;Dn−1;:::;D1;D0 = D, so that K ∼ K by a finite sequence of diagrams. Finally, suppose 0 0 00 0 0 K ∼ K and K ∼ K ; say that K is related to K by the sequence D = D0;D1;:::;Dn = D 0 00 0 0 0 0 00 and that K is related to K by the sequence D = D0;D1;:::;Dm = D . Then the sequence 0 0 0 00 00 D = D0;:::Dn;D0;D1;:::Dm = D is a finite sequence relating K and K , so ∼ is transi- tive. We may expand our definition of diagrams to oriented diagrams, where there is an assignment of orientation to the knot. The directed arc segments must give a consistent orientation of the knot. Equivalence of oriented diagrams is defined analogously to equivalence of unoriented diagrams. We may also define “knots” with more than one piece of string: Definition 1.5. A link is an embedding L :(S1)n ! R3 (where (S1)n is the product of S1 with itself n times) in which each component’s embedding is disjoint from the others. Intuitively, it is a finite collection of mutually disjoint closed loops of string in R3. Each “loop” of string is referred to as a component of a link. It is clear from the above definition that a knot is a one component link. Again, equivalence of links is defined similarly to equivalence of knots. Armed with a formal notion of what it means for two knots (or links) to be equivalent, one can begin to study the objects seriously. However, it can often be difficult to distinguish two knots from one another – from our definition, one must show there exists no sequence of Reidemeister 5 moves relating the two diagrams. Indeed, it may happen that two very similar diagrams represent completely different knots! Figure 1.4: Diagrams of two different knots. In Figure 1.4, for example, the diagrams differ at only one crossing – however, the diagram on the left is equivalent to the unknot while the diagram on the right is not. To aid in the study of knots, we need tools that can distinguish two knots from one another. These tools must illustrate properties which are shared across every possible diagram of a particular knot so that one can truly observe a difference between two different knots. In particular, these properties must remain unchanged by any sequence of Reidemeister moves. 1.2 KNOT INVARIANTS At first glance, one may be tempted to state that the number of crossings in a particular diagram is representative of the knot. However, Figure 1.3 presents a counterexample to that notion – the left diagram has four crossings, yet it represents the unknot, which is usually represented without crossings. Reidemeister move R1 adds one crossing to the diagram, while R2 subtracts two.
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