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CALIFORNIA STATE UNIVERSITY, NORTHRIDGE P-Coloring of Pretzel Knots A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics By Robert Ostrander December 2013 The thesis of Robert Ostrander is approved: |||||||||||||||||| |||||||| Dr. Alberto Candel Date |||||||||||||||||| |||||||| Dr. Terry Fuller Date |||||||||||||||||| |||||||| Dr. Magnhild Lien, Chair Date California State University, Northridge ii Dedications I dedicate this thesis to my family and friends for all the help and support they have given me. iii Acknowledgments iv Table of Contents Signature Page ii Dedications iii Acknowledgements iv Abstract vi Introduction 1 1 Definitions and Background 2 1.1 Knots . .2 1.1.1 Composition of knots . .4 1.1.2 Links . .5 1.1.3 Torus Knots . .6 1.1.4 Reidemeister Moves . .7 2 Properties of Knots 9 2.0.5 Knot Invariants . .9 3 p-Coloring of Pretzel Knots 19 3.0.6 Pretzel Knots . 19 3.0.7 (p1, p2, p3) Pretzel Knots . 23 3.0.8 Applications of Theorem 6 . 30 3.0.9 (p1, p2, p3, p4) Pretzel Knots . 31 Appendix 49 v Abstract P coloring of Pretzel Knots by Robert Ostrander Master of Science in Mathematics In this thesis we give a brief introduction to knot theory. We define knot invariants and give examples of different types of knot invariants which can be used to distinguish knots. We look at colorability of knots and generalize this to p-colorability. We focus on 3-strand pretzel knots and apply techniques of linear algebra to prove theorems about p-colorability of these knots. Lastly, we use the same techniques to prove similar results for 4-strand pretzel knots. vi Introduction In Chapter 1 we talk about basic concepts of knots and their definitions. We define knots both intuitively and mathematically and show how knots can be represented as projections onto a plane. In this thesis, we represent knots by their diagrams which are projections with added information at each crossing. We introduce the idea of "adding" two knots and from that define a prime knot. Also we define links and how they relate to knots. We describe specific types of knots, such as torus knots. Finally we define Reidemeister moves and describe how to apply them to knot diagrams. In Chapter 2 we define knot invariants, which are properties of knots. We ex- plain how to prove a property is a knot invariant. We introduce several types of knot invariants, including the crossing number of a knot, and the Alexander Polynomial. The focus of this chapter is to look at colorability of knots and showing that this is a knot invariant. We also give examples of how to compute certain invariants for specific knots. In Chapter 3 we define tangles and the addition of tangles. We define pretzel knots and look at their properties. We give examples of pretzel knots and apply previous definitions to this class of knots. We prove theorems on the equivalence of pretzel knots. We also find simple ways to work with knot invariants on pretzel knots. Finally we prove theorems about p-colorability of pretzel knots. 1 Chapter 1 Definitions and Background 1.1 Knots A mathematical knot is much like the concept of a knotted rope with both ends fused together. This definition gives us a very intuitive idea of what we are dealing with but in order to make a more accurate definition we use various topological definitions. First we introduce the idea of a homeomorphism in order to properly define a knot. This allows us to give a mathematical foundation to the intuitive explanation behind a knot. Definition 1 (Homeomorphism) A homeomorphism is a continuous bijection such that its inverse is continuous. Definition 2 (Knot) A Knot is a homeomorphic image of S1 in R3 Given two knots it is important to be able to distinguish the knots. This is the major focus of this paper. From the definition of homomorphism we give a unique name to the knot that is formed from the identity on S1, we call it the unknot. Next we expand upon the idea of two knots being the same. Definition 3 ( Ambient Isotopy) An Ambient Isotopy is a continuous distortion in 3 R . Given knots K1 and K2, an ambient isotopy from K1 to K2 is a continuous 3 3 function F: R X [0,1] ! R such that F(K1, 0) = K1 and F(K1, 1) = K2 and 8 t 2 [0,1] Ft is a homeomorphism. Definition 4 (Knot Equivalence) Two Knots are equivalent if there is an ambient isotopy between the two knots. When examining knots it is difficult to deal with the embedding of the knot in R3. In order to simplify this problem we look at the projection of a knot onto a plane. This can lead to a few problems which can be solved by rotating the original knot slightly. The figure below illustrates one of the problems and a solution when projecting a knot onto a plane. 2 Also, when projecting onto a plane important information about the knot is lost. After projecting onto a plane the over-crossing and under-crossing can not be distin- guished. We modify the projection slightly in order to retain this information. In order to accomplish this we remove a portion of the under-crossing so that it can be distinguished from the over-crossing. This is further illustrated in the figure below. Definition 5 (Knot Diagram) Given a knot, any projection onto the plane such that there is at most double points and at each crossing part of the under crossing is removed is called a knot diagram. Once we have a knot diagram we can orient the knot. This involves choosing a di- rection to go around the knot. It is easy to see that given a knot there are only two possible orientations for it. The figures below shows the two possible types of crossing for an oriented knot, the right-hand and left-hand crossings. Right-handed crossing Left-handed crossing 3 Example 1 The figures below are two diagrams of the trefoil knot. The trefoil knot is the only non-trivial knot that can be represented as a three crossing knot. The example above shows the trefoil knot and its mirror image. At first glance it is hard to see that they are different knots and harder to prove that they are not equivalent knots. A knot that is not equivalent to its mirror image is called a chiral knot, otherwise it is achiral. A knot is called invertible if given both possible orientations of a knot, they are equivalent. 1.1.1 Composition of knots If you have knot diagrams of two knots it is possible to define the composition of the two knots. After removing a small arc from each knot and connect each endpoint to an endpoint on the other knot we obtain a new knot. The standard notation for the composition of two knots K1 and K2 is K1 # K2. We assume that the arc chosen on each knot is on the outermost strand for each so that no new crossings are formed. This is illustrated in the figure below. K1 K2 K1#K2 4 If you cannot express a knot as the composition of two nontrivial knots it is called a prime knot. A non-prime knot is called a composite knot. This leads to the ques- tion can you ever get the trivial knot from the composition of two non trivial knots? This is an important question because if it was possible then every knot would be a composite knot which would remove the usefulness of this definition. It can be shown that the composition of two non-trivial knots can not be the unknot. Hence the set of knots is not a group under the operation #. This is outside the scope of this paper so we will take it as fact. Also note that all of the knots in the table at the end of this paper are prime knots. As one would think, the idea of prime knots parallels the idea of prime numbers and as with positive integers we can factor composite knots into the composition of prime knots. Unlike integers we can not say that composing two knots two different ways will give us the same knot. In order to fix this we will first need to give both knots an orientation. Given two oriented knots, removing a small arc from both knots and connecting them in such a way that the orientation of the two knots match will always give us the same knot. This leads to the idea that given two knots if one of the knots is invertible then the composition of them will always be the same knot[1]. 1.1.2 Links Thinking of knots as closed curves in R3 we can also think of multiple knots in R3. An n-component link is a set of n homeomorphic images of S1 in R3 such that none of them intersect each other. Note that a knot is a one-component link. A link is called splittable if the components of a link can be deformed so that they are on different sides of a plane in R3. A link is called Brunnian if the link itself is nontrivial, but if any of the components are taken away you get the trivial unlink. Definition 6 (Unlink) A link is said to be the un-link of n components if it is split- table and each individual component is equivalent to the unknot.
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