THE ADDITIVITY OF CROSSING NUMBER WITH RESPECT TO THE COMPOSITION OF KNOTS JASON GRANDY SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERS OF SCIENCE IN MATHEMATICS NIPISSING UNIVERSITY SCHOOL OF GRADUATE STUDIES NORTH BAY, ONTARIO © Jason Grandy August 2010 I hereby declare that I am the sole author of this Thesis or major Research Paper. I authorize Nipissing University to lend this thesis or Major Research Paper to other institutions or individuals for the purpose of scholarly research. I further authorize Nipissing University to reproduce this thesis or dissertation by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. iii Abstract This paper will investigate the additivity of the crossing number with respect to the composition of knots. The additivity of the crossing number is a long standing conjecture. The paper presents proofs of this conjecture for alternating knots and torus knots. For alternating knots, the paper uses the Jones Polynomial to show the alternating diagram has minimal degree, and proves the composition of two alternating knots is another alternating knot. For torus knots, the paper’s main ingredient is a closed form equality for the crossing number involving the braid index and genus of the knot. We then show the additivity under composition of these components of the formula to prove the additivity of the crossing number. iv Contents Abstract ................................................................................................................................................. iii Introduction ............................................................................................................................................ 1 Knots and Knot Diagrams ....................................................................................................................... 1 Crossing Number .................................................................................................................................... 2 Knot Diagram Regions ............................................................................................................................ 3 Reidemeister Moves ............................................................................................................................... 4 Composition of Knots ............................................................................................................................. 5 Braids ...................................................................................................................................................... 7 Bridges .................................................................................................................................................... 8 Writhe Number .................................................................................................................................... 10 Torus Knots ........................................................................................................................................... 10 Alternating Knots .................................................................................................................................. 16 Seifert Surfaces ..................................................................................................................................... 17 A Closed Form Equation for the Crossing Number............................................................................... 19 Knot Genus ........................................................................................................................................... 21 The Bracket Polynomial and Jones Polynomial .................................................................................... 23 Span of the Bracket Polynomial and Jones Polynomial ........................................................................ 28 Main Conjecture ................................................................................................................................... 31 Additivity of Composition Operation for Torus Knots .......................................................................... 32 Additivity of Composition Operation for Alternating Knots ................................................................. 33 References ............................................................................................................................................ 33 v 1 Introduction The goal of this paper is to present the findings concerning the additivity of the crossing number with respect to the composition operation of knots. That is, for two knots A and B, ( ) ( ) ( ) where c(A) is the crossing number for the knot A, and “#” is the composition operation. One direction of the equality requires just an observation, of two knot diagrams, so we can see ( ) ( ) ( ). This conjecture has been solved for alternating knots by Murasugi [4], and a family of knots that includes torus knots by Gruber [1]. As well, bounds have been placed on the other inequality by Lackenby [2]. He proved the inequality ( ) ( ( ) ( )). The field of knot theory is still very young, with a lot of major progress happening within half a century such as the Jones Polynomial, and the two main conjecture proofs given. Definitions and concepts shown are easy to explain and visualize, and often so are the theorems. Nevertheless the theorems require advanced tools to prove. The main conjecture presented follows this trend with the simplicity that it can be explained to someone with no post secondary mathematics education, yet still remains unsolved. The findings will be presented in a way that a person with no knowledge of knots will be able to find all information needed to understand the concepts in the paper itself. In short, this paper will be self contained. Knots and Knot Diagrams Definition 1. A knot is an embedding of S1 in R3. A link is one or more knots with no intersections. For the scope of this paper, we will only consider smooth or piece-wise linear embeddings of S1 to R3. 1 2 Definition 2. Two links are equivalent if there exists an orientations preserving diffeomorphism or, depending on situations and knots we consider, a piece-wise linear homeomorphism of R3 on itself that maps one link onto the other. Definition 3. A link diagram is projection of a link onto a plane such that the projection does not cross itself more than twice at any point. Since a link in R3 doesn’t intersect itself, any intersections on a link diagram are labelled with overcrossings and undercrossings to represent which section of the link lies above and which lies below. Definition 4. Two link diagrams are equivalent if the links they are projections of are equivalent. For several properties it is needed to give a link an orientation. This will be represented by an arrow on a link diagram for several examples in this paper. The following terminology will be used throughout the paper. Definition 5. A string is a connected subset of a knot. A strand is a string on a knot diagram whose endpoints begin and end at crossings. Definition 6. A splitting point is a crossing that, if deleted, would split the knot into two. Crossing Number Definition 7. The crossing number of a link diagram, denoted c(D) for a diagram D of a link K, is the number of times the link crosses over itself. The crossing number of a link, denoted c(K), is the minimum crossing number of all its link diagrams. 2 3 Definition 8. Any link with a finite number of crossings is called a tame link, and any link with an infinite number of crossings is called a wild link. Aside from one brief proof, all links considered in this paper will be tame links. Moreover, we will usually assume all links are piece-wise linear or smooth. The only knot with zero crossings is the trivial knot, which also called the unknot. The only two knots with three crossings are the trefoil knot, and its mirror image. The knot with four crossings is the figure eight knot. These three knots will be used extensively in this paper. Knot Diagram Regions Definition 9. A region is a connected component of the complement of a link diagram in the plane. Each lettered area in the picture below is a region. 3 4 If we turn a knot’s crossings into vertices and the knot itself into an edge, we can create a graph where the faces are the regions. By using the Euler Characteristic, we know where V is the number of vertices, E is the number of edges, and F is the number of faces. We notice that is incident with 2 vertices, and each vertex is incident with 4 edges. This implies that there are twice as many edges as vertices, so , and so . Therefore, including the outside face, there are 2 more regions than crossings in any given knot. Thus, we proved the following proposition. Proposition 10. The number of regions of a knot diagram equals ( ) . Reidemeister Moves The Reidemeister Moves are a set of moves which can transform a link diagram into and only into an equivalent link diagram. These moves are easily described and visualized. The diagrams given also hold for all symmetries possible. Reidemeister Move 1: To add or remove a loop Reidemeister Move 2: To pass one string over or under another string, or the reverse Reidemeister Move 3: To pass a string over or under a crossing 4 5 Theorem 11 ([3] Th 2.1). Two link diagrams are equivalent if and only if there exists a series of Reidemeister moves transforming