Eindhoven University of Technology BACHELOR Knots and codes Nijsten, Anne I.O. Award date: 2019 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain Knots and codes Anne Nijsten Supervisor: dr. G.R. Pellikaan Eindhoven University of Technology Department of Mathematics and Computer Science July 19, 2019 Abstract Since the 1860s, interest in mathematical knot theory has been risen, partly due to its applications in other disciplines, like molecular biology and quantum physics. The central problem in this field is determining whether two knots are equivalent. This report gives an overview of knot theory and some of its applications. Further, after introducing some concepts of coding theory, it presents the idea of regarding knots as codes and using properties of codes to discover more about knots. 2 Contents 1 Introduction 4 2 Knot theory 4 2.1 Applications . .5 2.2 Knots . .6 2.3 Braids . .9 2.4 Colorability . 10 2.5 Number of colorings . 12 2.6 Knot determinant . 14 2.7 Alexander polynomial . 14 3 Coding theory 15 3.1 Applications . 16 4 Colorings and codes 17 4.1 Reidemeister moves . 17 4.2 Kauffman-Harary conjecture . 17 4.3 Coloring of braids . 18 5 Future Research 18 6 Conclusion 19 7 References 20 3 1 Introduction Knots have been around since prehistoric times, and are still part of everyday life in many ways. Practical uses include tied shoelaces and are used by climbers and sailors to fasten objects. The Celts used them in an artistic way as decoration, nowadays children knot friendship bracelets as accessories. Incas used them to store information on trading as so-called quipu, or talking knots. Also in mathematics, knots have been around for about 150 years. Central in mathematical knot theory is classification of knots by determining whether two knots are equivalent. This can be done by using techniques from other fields within mathematics. In this report, the theory of error correcting codes will be used to describe knots. Knowledge about knots can be used in many other fields, like physics, chemistry and molecular biology, as also described later in this report. In section 2 the theory of mathematical knots and braids is introduced. There, the concept of knot equivalence and knot invariants are explained, along with some examples of invariants, like `-colorability, knot determinant and Alexander polynomial. Also, some applications of knot theory are given. Section 3 introduces some terms from coding theory, like codes and their equivalence. The theory of knots and codes will be combined in section 4. There, colorings knots and braids are regarded as codes. The final sections discuss possible continuations for future work and conclude on the work described in this report about the combination of knots and codes. 2 Knot theory In mathematics, a knot is a simple, closed curve in space. That is, a knot is an embedding of a unit circle S1 in three-dimensional Euclidean space R3, given by an injective, continuous function f : S1 ! R3 that yields an homeomorphism between S1 and f(S1) = K. Now S1 and K being homeomorphic means that there exists a continuous function between S1 and K that has a continuous inverse. When a knot is deformed without passing through itself, the resulting knot is equivalent. More 3 3 formally, two knots K1 and K2 are equivalent if there exists an homeomorphism h : R ! R , such that h(K1) = K2. These definitions allow us to distinguish between two types of knots: tame knots an wild knots. A knot is tame if it is equivalent to a polygonal knot, that is, a knot that can be represented by a closed chain of a finite number of connected line segments in R3. A knot that is not tame is called a wild knot. In the following text, a knot is assumed to be a tame knot, unless otherwise stated. Examples of (tame) knots are the unknot, or trivial knot, and the trefoil knot. To visualize and manipulate knots, they can be projected on a plane, resulting in knot diagrams like the ones in figure 1 when pictured. The separate lines in these diagrams are called strands. The areas between strands and outside the diagram are called regions. At points where a projection of a knot on a plane crosses itself, in a knot diagram the overstrand is distinguished from the understrand by drawing a break in the understrand. In a regular projection of a knot onto the plane, no more than two points of the knot project to the same point on the plane, the amount of points where the projected knot crosses itself is finite, and the knot crosses itself only transversally. Lemma 2.1. Let D be a diagram of a knot K with n crossings. Then there are n + 2 regions in the diagram. Proof. Construct a graph from D by representing crossings by vertices, and strands by edges, where overstrands in crossings are represented by two edges. Then we have a planar graph, of which each vertex has degree 4. This graph has as many regions as D has. By the handshaking 1 Pn 1 Pn lemma, the graph has 2 1 d(v) = 2 1 4 = 2n edges. Then, by Euler's formula, the graph has 2n − n + 2 = n + 2 regions. In figure 2, a polygonal knot is depicted, which is equivalent to the trefoil knot in the center of figure 1. A wild knot is depicted on the right in figure 2. There the dots indicate an infinite 4 amount of overhand knots following the 4 already drawn. Note that an overhand knot by itself, which is a trefoil knot of which one of the strands is cut through, is not a mathematical knot. The equivalence of two knots may be hard to determine, e.g. in figure 1 it is not immediately clear that the left and right knot diagrams are equivalent. Determining the equivalence of two knots is the central problem in knot theory. [1] Figure 1: The trivial knot (left), trefoil knot (center) and another projection of the trivial knot Figure 2: A polygonal knot (left) and a wild knot A disjoint union of knots, which may be tangled up or linked together, is called a link. More formally, L is a link if it is homeomorphic with a disjoint union of one or more circles. Similar to knots, links can be pictured by link diagrams. A splittable link is a link that consists of several disjoint knots, which are called components of a link. The three knots in figure 1 together could be regarded as a splittable link with three components, because it consists of several knots that can be separated in the plane. The unlink with n components is the splittable link consisting of n disjoint unknots, and is depicted for n = 2 in figure 3. Examples of nontrivial links are the Hopf link and Borromean rings, which consist of two and three components, all unknots, respectively. In figure 3 their link diagrams are pictured. When links can be transformed into each other by deforming their components without letting them pass through themselves or each other, they are considered equivalent. Figure 3: The trivial link of 2 components (left), Hopf link (center) and Borromean rings 2.1 Applications Knots have had many applications over the years. In everyday use they can, for example, be found in shoelaces, as macram´edecorations, aboard sailing ships or in climbing ropes. Apart from fastening physical objects, they have also been used for record keeping, like in South-American quipu, or \talking knots", a coding system used by the Inca people in Peru [6]. These knots often differ from mathematical knots, as the ends of a cord in which the knot is placed are usually not connected. The interest in studying mathematical knots resulted from the hypothesis of Lord Kelvin that atoms were knotted vortices in ether, presented in 1867 [20]. With the idea that different materials 5 consisted of different knots and links, and that their characteristics could be explained by these compositions, the listing of unique knots was started. Although the theory of vortex atoms had become obsolete around the beginning of the twentieth century, investigation of knots continued. Currently, applications of mathematical knots can for example be found in biology, chemistry and physics. In molecular biology, knot theory can be used to model DNA molecules and to study how these are untangled in an efficient way by enzymes [19]. In chemistry, knots are used to determine chirality of molecules, that is, whether the molecule can be deformed into its mirror image [12]. In physics, braids, which are closely related to knots, are used to form logic gates in the idea of a topological quantum computer, which has a lower error rate than other quantum computers that use trapped quantum particles.