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On the Additivity of Crossing Numbers ON THE ADDITIVITY OF CROSSING NUMBERS A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Alicia Arrua 2015 SIGNATURE PAGE THESIS: ON THE ADDITIVITY OF CROSSING NUMBERS AUTHOR: Alicia Arrua DATE SUBMITTED: Spring 2015 Mathematics and Statistics Department Dr. Robin Wilson Thesis Committee Chair Mathematics & Statistics Dr. Greisy Winicki-Landman Mathematics & Statistics Dr. Berit Givens Mathematics & Statistics ii ACKNOWLEDGMENTS This thesis would not have been possible without the invaluable knowledge and guidance from Dr. Robin Wilson. His support throughout this entire experience has been amazing and incredibly helpful. I’d also like to thank Dr. Greisy Winicki- Landman and Dr. Berit Givens for being a part of my thesis committee and offering their support. I’d also like to thank my family for putting up with my late nights of work and motivating me when I needed it. Lastly, thank you to the wonderful friends I’ve made during my time at Cal Poly Pomona, their humor and encouragement aided me more than they know. iii ABSTRACT The additivity of crossing numbers over a composition of links has been an open problem for over one hundred years. It has been proved that the crossing number over alternating links is additive independently in 1987 by Louis Kauffman, Kunio Murasugi, and Morwen Thistlethwaite. Further, Yuanan Diao and Hermann Gru­ ber independently proved that the crossing number is additive over a composition of torus links. In order to investigate the additivity of crossing numbers over a composition of a different class of links, we introduce a tool called the deficiency of a link. When the deficiency is equal to 0, we are able to use a powerful result to prove that the crossing number is additive over a composition of links with de­ ficiency 0. Applying this result, we are able to focus on computing the deficiency of a class of links called Montesinos links, specifically alternating pretzel knots and non-alternating pretzel knots. iv Contents Acknowledgements iii Abstract iv List of Figures ix 1 Introduction 1 1.1 Historical Background ......................... 1 1.2 Knot Theory Background ....................... 3 1.3 Problem Statement ........................... 10 2 On Surfaces and Polynomials 11 2.1 Surfaces and Knots ........................... 11 2.2 Seifert Surfaces and Genus ....................... 14 2.3 Polynomials ............................... 23 3 Additivity of Crossing Numbers 30 3.1 Preliminaries .............................. 30 3.2 Alternating Knots ............................ 37 3.3 Torus Knots ............................... 41 v 4 Pretzel Knots 46 4.1 Tangles ................................. 46 4.2 Montesinos Links and Pretzel Knots . 48 4.3 Nonalternating Pretzel Links ...................... 55 4.4 Future Work ............................... 60 Bibiliography 61 vi List of Figures 1.1 Some examples of knots and links ................... 4 1.2 A projection ............................... 4 1.3 A diagram ................................ 4 1.4 Regular crossings ............................ 5 1.5 Nonregular Crossings .......................... 5 1.6 Type I Reidemeister moves ....................... 6 1.7 Type II Reidemeister moves ...................... 6 1.8 Type III Reidemeister moves ...................... 7 1.9 Perko pair ................................ 8 1.10 Examples of knots with orientation . 8 1.11 Composition of knots .......................... 9 1.12 Composite knot ............................. 9 2.1 Deformation from a sphere to a cube . 11 2.2 Surfaces with 1, 1, and 3 boundary components . 12 2.3 Possible intersections of triangles . 12 2.4 Sphere and torus ............................ 13 2.5 Assigning an orientation ........................ 15 2.6 Eliminating the crossings ........................ 15 vii 2.7 Two views of the Seifert circles produced . 15 2.8 Two views of the Seifert surface produced . 16 2.9 A band joining two Seifert circles ................... 16 2.10 Adding vertices, edges, and faces to a band . 16 2.11 Edges for each band .......................... 17 2.12 Creating a composite knot from K1 and K2 ............. 20 2.13 Removing a point of intersection between F and S ......... 21 2.14 Intersection loops on S ......................... 21 2.15 Surgery along disk D .......................... 22 2.16 +1 crossing ............................... 25 2.17 −1 crossing ............................... 25 2.18 Hopf link with orientation ....................... 26 2.19 Labeling the regions of the projection plane . 26 2.20 B-split and A-split ........................... 27 2.21 Trefoil knot with labeled crossings . 28 2.22 All states of the trefoil knot ...................... 28 3.1 2-bridge knots .............................. 31 3.2 A braid and its closure ......................... 32 3.3 A meridian curve and a longitude curve on a torus . 41 3.4 Two views of the trefoil knot embedded on a torus . 41 3.5 Standard representation D of T .................... 43 4.1 Example of a tangle ........................... 46 4.2 Another example of a tangle ...................... 46 4.3 The 0 tangle ............................... 47 viii 4.4 The 1 tangle .............................. 47 4.5 A Montesinos knot ........................... 49 4.6 A diagram of the knot Kn ....................... 50 4.7 Condition 1 ............................... 51 4.8 Condition 2 ............................... 51 4.9 A labeling of the trefoil knot ...................... 51 4.10 A labeling of the 74 knot ........................ 51 4.11 One possibility of ri ........................... 52 4.12 Another possibility of ri ........................ 52 4.13 The figure eight knot with relators . 52 4.14 A pretzel knot (3; 3; :::; 3; 2) with labelings from Sn ......... 54 3 4.15 K2 = (3; −3; 2) ............................. 56 3 4.16 Two views of the Seifert surface of K2 ................ 56 3 4.17 Kn = (3; −3; 3; :::; 3; 2)......................... 57 2k+1 4.18 Kn = (2k + 1; −(2k + 1); 2k + 1; ::: 2k + 1; 2) . 59 ix Chapter 1 Introduction 1.1 Historical Background Knot theory is a particularly interesting field of mathematics in which questions are easily stated and not so easily solved. One of the earliest known works came from Carl Friedrich Gauss whose notes involving a collection of knot drawings from 1794 were found and whose interest in the subject continued, later deriving the “Gauss Linking Number” in 1833. After some time, physicists and chemists also became interested in knot theory and in the 1860’s, Sir William Thomson (Lord Kelvin), an English physicist, attempted to explain the different types of matter by hypoth­ esizing that different knots would correspond to different elements. He did this as an attempt to combine two ideas which were prevalent in the scientific society: the idea that matter is composed of atoms and the idea that matter consisted of waves. It was after this that Scottish physicist Peter Guthrie Tait began tabulating knots in 1867 in order to classify knots. In 1885, he published tables of knots up to ten crossings. Independently, American mathematician Charles Newton Little also 1 began tabulating knots and in 1899 he published tables of nonalternating knots up to ten crossings and deduced that there were 43 such knots. However, in 1974, Kenneth Perko discovered that two of the knots in Little’s table were actually the same knot so there were really 42 nonalternating knots of ten crossings or less [1]. Right around the time that the tables were published, Thomson’s theories were shown to be incorrect. However, knot theory continued to be pursued as a math­ ematical field and Tait’s interest in knots also continued as he investigated the properties of reduced knot diagrams. As a consequence, he produced three impor­ tant conjectures which continue to be useful and will be stated in Chapter 2 [1]. He provided nonrigorous proofs for these conjectures and it wasn’t until the 1980’s that these were rigorously proven after the discovery of the Jones polynomial. The need to distinguish between knots led mathematicians to discover new in­ variants which could then be used in rigorous proofs. Of these invariants, Laurent polynomials are particularly interesting in that a unique Laurent polynomial of one variable can be computed from a projection of a knot, even if it is computed using different projections of the knot. There are several types of interrelated Laurent polynomials that can arise from a knot. American mathematician James Waddell Alexander II first produced a polynomial associated to knots and links in 1928 called the Alexander polynomial. This polynomial was the only known polynomial invariant until 1984 when a mathematician from New Zealand, Vaughan Jones, dis­ covered a new polynomial that is now known as the Jones polynomial. Within four months, a two-variable generalized Jones polynomial called the HOMFLY polyno­ mial was discovered by mathematicians Jim Hoste, Adrian Ocneau, Kenneth Millet, Peter J. Freyd, William B. R. Lickorish, and David N. Yetter and independently by J´ozef H. Przytycki and Pawell Traczyk. The Jones polynomial will be discussed 2 in further detail in Chapter 2. These polynomials, along with other invariants, allow us to examine and classify knots in order to discover and work with the properties which arise from each type of knot. More details related to the history of knot theory can be found in [1] and [12]. In the rest of this chapter,
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