Eindhoven University of Technology

BACHELOR

Knots and codes

Nijsten, Anne I.O.

Award date: 2019

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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. This is because small perturbations in the environment of a quantum particle will cause it to loose its state, while such perturbations will not affect the topological state of a quantum braid, thus these are more stable [9].

2.2 Knots In 1927, it was proven by Kurt Reidemeister that by using a sequence consisting of three types of moves, any two regular projections of equivalent knots can be transformed into each other [16]. These so-called Reidemeister moves are depicted in figure 4. A way to deform the unknot into the rightmost knot in figure 1 using the Reidemeister moves is shown in figure 5.

Figure 4: The type I (left), II (center) and III Reidemeister moves

II I III II

Figure 5: The deformation of the unknot using Reidemeister moves.

A knot invariant f is a function that assigns to a knot K an object f(K) such that for equivalent knots K1 and K2 you have that f(K1) = f(K2), and thus can be used to distinguish (some) knots [17]. An example is the crossing number of a knot, which is the minimum number of crossings of any knot diagram depicting the knot. The unknot has crossing number 0 and the trefoil knot has crossing number 3. Two knots K1 and K2 can be added up, or composed, resulting in the knot sum K1#K2. This operation is performed like depicted in figure 6 by taking away two arcs in strands of each knot and connecting the resulting endpoints. The connecting arcs may not cross each other or any other strands of the knots. The knot sum of any knot K with the unknot results again in K.

6 + =

Figure 6: The composition of two trefoil knots into the granny knot.

However, this operation is not well-defined: the resulting knot of a knot sum may depend on the position of the arcs that are removed in order to connect the knots. To solve this, the orientation of a knot is introduced, which is a direction to travel around the knot. In a knot diagram, the orientation can be marked by placing arrows coherent with the direction of travel on the knot. Two oriented knots K1 and K2 are equivalent if there exists an orientation-preserving 3 3 homeomorphism h : R → R , such that h(K1) = K2. The Reidemeister moves preserve the orientation of a knot, as can be seen by placing arrows on the strands shown in figure 4. If a knot K with clockwise orientation is equivalent to the K with counterclockwise orientation, it is called invertible. The trefoil knot is an example of an invertible knot. This can be seen by taking a trefoil knot and giving it an orientation. If you turn the knot over, as shown in figure 7, you again have a trefoil knot, but with the opposite orientation.

180°

Figure 7: Turning over an oriented trefoil knot gives a trefoil knot with opposite orientation

Two oriented knots can be added in two ways: they either match in orientation or not. This is depicted in figure 8, where in both cases the left diagram is replaced by the right diagram in order to perform the knot sum. Any sum of the knots K1 and K2 where the orientations match up will result in the same knot. To see why, when you have any two knot sums of K1 and K2 with matching orientation, you can take one and slide K1 along K2 in the direction of its orientation until you get the other one. Also, any sum of the knots K1 and K2 where the orientations differ will result in the same knot, which may be different from the knot that results from adding them with matching orientation. From this we find that the knot sum of two unoriented knots can result in no more than two different knots.

K1#K2 K1#K2

K K K1 2 K1 2

Figure 8: Matching orientation (left) and differing orientation

In figure 9 the knot sum of the trefoil knot with its mirror image is depicted. This results in the so-called square knot, which is different from the granny knot in figure 6. The trefoil knot is a chiral knot, which means that it is not equivalent to its mirror image. An amphichiral knot is equivalent to its mirror image. The figure-8 knot is an example of an amphichiral knot. In figure 10 this knot and its mirror image are pictured, along with a series of Reidemeister moves to show their equivalence..

7 + =

Figure 9: The composition of a trefoil knot and its mirror image into the square knot.

3x III 2x II III 2x II 1x II

2x II

2x I II

Figure 10: The figure-8 knot is equivalent to its mirror image.

One type of knots are the prime knots, which are knots that are nontrivial and cannot be written as the sum of two nontrivial knots. A common notation for these knots is nk, where n is the number of crossings for the diagram as drawn in the list of prime knots in [17] and k is an ordering number to distinguish prime knots that have a projection with the same number of crossings. As an example, the trefoil knot is denoted by 31 and the figure-8 knot by 41. In this notation, no distinction is made between chiral knots and their mirror images. One specific type of prime knots are the torus knots. Take p and q with gcd(p, q) = 1. A (p, q)-torus knot crosses a meridian of an unknotted torus q times and a longitude closed curve on the surface of a torus p times [1]. That is, it wraps q times around the axis of revolution and loops p times through the hole of the torus. If gcd(p, q) = d > 1, the result is a torus link, consisting of d components. For visualisation, a meridian and longitude curve of a torus are depicted in figure 11. As an example the trefoil knot can be taken, as it is a (3, 2)-torus knot, also denoted as T (3, 2). In figure 11 the trefoil knot embedded on a torus is pictured. An example of a torus link with two components is T (2, 2), which is also known as the Hopf link.

Figure 11: A torus with a meridian in red, longitude in blue and an axis of revolution in green (left) and a torus with trefoil knot in blue (right)

A knot diagram is called reduced if it does not contain crossings of the form shown in figure 12. An alternating knot is a knot for which a diagram exists in which you have alternating understrands and overstrands when the knot is followed in a fixed direction. Examples of alternating knots are the trefoil knot and (2, n)-torus knots.

8 The number of crossings in any reduced alternating diagram of an alternating knot is equal to the crossing number of the knot. This statement results from three conjectures made by mathematician Peter Tait in the 19th century after he started tabulating knots in response to Lord Kelvin’s ideas about atoms being knotted vortices. These conjectures were not proven until the late 1980s [1].

D1 D2 D1 D2

Figure 12: Crossings that may not occur in a reduced diagram, where D1 and D2 form the diagram apart from the crossing

2.3 Braids A braid is a set of k strings, connecting two horizontal bars in such a way that each string intersects a horizontal plane, placed between the bars, exactly once. Some examples of braids can be seen in figure 13.

Figure 13: Some braids with 2 or 3 strings

Two braids are equivalent when the strings of one of them can be rearranged without passing through themselves or each other in such a way it looks like the other braid. In figure 13, the two leftmost braids are equivalent: they can be transformed into each other using a type II Reidemeister move. The three rightmost braids are all different from each other.

Figure 14: Some closed braids with 2 or 3 strings

To relate knots, links and braids, the closure of a braid is used. A braid is closed by connecting corresponding pairs of ends of the strings. That is, the ends leftmost ends will be connected to each other, the ends second from the left will be connected to each other, etc. The closures then form knots or links. In figure 14, the closures of the first, third and fourth braid in figure 13 are pictured. They correspond to the Hopf link, trefoil knot and figure-8 knot respectively. In particular, Alexander’s theorem states that every knot or link is a closed braid, as proven by James Alexander in 1923 [2]. Two braids b1 and b2 with k strings can be concatenated, by joining the ends of b1 to the start of b2, as shown in figure 15, resulting in a braid b1b2 with k strings. It can easily be seen that this operation is associative. The braid group on k strings Bk is the group with equivalence classes of braids with k strings as elements and the previously described concatenation of braids as group operation. The identity element of Bk is the unbraid with k strings, represented by k parallel strings. The inverse of a braid is given by mirroring the braid on one of its horizontal bars.

9 b b b1 1 2 b1

mirror axis

-1 b2 b1

Figure 15: Concatenation of braids (left), unbraid with 3 strings (center) and a braid and its inverse

A way to describe braids without drawing them is by using braid generators, or elementary th braids. The elementary braid σi denotes a braid with k strings in which the i string passes under th −1 th the (i+1) string. The elementary braid σi denotes a braid with k strings in which the (i+1) string passes under the ith string. Now a braid with k strings can be described by a sequence of concatenated elementary braids, also called a braid word. As an example, the braid on the left in −1 −1 −1 −1 figure 10 can be described by σ1 σ1 and the braid on the right by σ1 σ2σ1 σ2. A torus link q T (p, q) can be described as the closure of the braid (σ1σ2 . . . σp−1) [11]. As stated by Artin [5], the braid group Bk then is described by the generators σ1, ..., σk and the following relations:

σiσj = σjσi, |i − j| > 1 (1) −1 σiσi = 1, i = 1, ..., k − 1, and (2) σiσi+1σi = σi+1σiσi+1, i = 1, ..., k − 2. (3)

In figure 16, these relations are pictured, it can be seen that they describe equivalences between braids. The first of them is also called far commutativity. As can be seen, the second and third relation correspond to the type II and type III Reidemeister moves respectively. Note that the result of performing a type I Reidemeister move on a braid is not a braid.

σ σi i σi+1 σi+1 σi σi+2

σi+2 σi σ σi i -1 σ σi i+1

Figure 16: Relations between braids

2.4 Colorability Tricolorability is a knot invariant that describes whether a knot can be nontrivially colored using three colors, where at each crossing the strands either have three different colors or all have the same color. Here nontrivially colored means that more than one color is used when coloring the knot. The invariant was introduced by R. Fox around 1960 as an easy to understand knot invariant [15]. In figure 17 it is shown that each of the three Reidemeister moves can be made without affecting tricolorability. As tricolorability is invariant under the Reidemeister moves, it can be concluded that, when any diagram of a certain knot is (not) tricolorable, all the diagrams of that knot are (not) tricolorable.

10 As an example we see that the unknot only has trivial three-colorings and the trefoil knot has a three-coloring (e.g. when each strand in the diagram of the trefoil knot in figure 1 is given a different color). Therefore, the unknot and trefoil knot are different knots.

Figure 17: Tricolorability of Reidemeister moves

When the elements in Z3 are used for a three-coloring, the solutions can be found using linear algebra, modulo 3. The equation x1 + x2 + x3 ≡ 0 mod 3 holds if and only if either x1 = x2 = x3 or x1, x2 and x3 are pairwise distinct. Therefore, we can label each of the n strands of a knot diagram as xi, where xi ∈ 0, 1, 2, and for every crossing in the diagram, like the one in figure 5, the equation xi + xj + xk ≡ 0 mod 3 has to hold [7]. For a knot with n crossings this leads to a system of n equations, and all solutions of this system denote a coloring. For each knot, there are at least 3 solutions, namely the trivial solutions, where all strands have the same color.

xi xk

xj

Figure 18: Marking of strands of a crossing

The concept of tricolorability can be generalized to `-colorability, which is also invariant under the Reidemeister moves [8]. This is done using the equation xi + xj ≡ 2xk mod 3, which is equivalent to xi + xj + xk ≡ 0 mod 3. In the general case, the elements of Z` can be used as colors for the strands. For each crossing of an `-colorable knot, the equation xi + xj ≡ 2xk mod ` must hold. The solution space of the system of homogeneous equations of a knot K with n n strands is a subspace of Z` . When ` is prime, the solution space is a vector space with dimension dim`(K). When this dimension is greater than or equal to 2 then the knot K is called nontrivially `-colorable. For a link to be `-colorable, the same rules must hold. In figure 19, the invariance of an `-coloring is pictured. With the Reidemeister moves, trivial colorings go over to trivial colorings, nontrivial colorings go over to nontrivial colorings.

x x x x x x x x x x x1 x1 1 2 1 2 1 2 3 1 2 3

2x2-x1 2x1-x2 2x3-x2

2x3-x1 2x3-x1 x =x 2 1 x1

2x3-2x1+x2 2x3-2x1+x2

Figure 19: `-colorability of Reidemeister moves

11 2.5 Number of colorings `-colorability is binary invariant: a knot is either `-colorable or it is not, which makes it rela- tively weak. To make it stronger, the number of `-colorings of a knot K, denoted by col`(K), is introduced. This number is also preserved under the Reidemeister moves, as each `-coloring is preserved under these moves, so it is a knot invariant [15].

dimp(K) Proposition 2.2. If p is a prime number, then we have that colp(K) = p .

Proof. For an p-coloring of a knot K, the colors can be denoted as elements of Zp. All colorings n of a certain diagram of K with n arcs that use at most p colors can be denoted as elements of Zp , which has pn elements. When p is prime, the allowed p-colorings of K can be found by solving n a homogeneous system of linear equations with n variables, resulting in a linear subspace of Zp with dimension dimp(K). Therefore, it has a basis with dimp(K) elements. As there can be made dimp(K) dimp(K) p colorings, which are linear combinations of this basis, we find that colp(K) = p .

n n Note that col`(K) ≤ ` , the number of elements of Z` and col`(K) ≥ `, the number of trivial colorings. The number of colorings for a diagram can for example be found by solving the system of linear equations described in section 2.4. Then the number of solutions is the number of colorings.

x1 x1 x x 3 2 x4

x3 x2

Figure 20: Marked trefoil (left) and figure-8 knot

When we row reduce the coefficient matrices corresponding to the systems of equations of the diagrams of the trefoil knot and figure-8 knot in figure 20 modulo 3, we obtain the following matrices: −2 0 1 1  1 0 0 2 −2 1 1  1 1 1 1 −2 0 1 0 1 0 2 1 −2 1 ∼ 0 0 0 and   ∼   .      1 1 −2 0  0 0 1 2 1 1 −2 0 0 0     0 1 1 −2 0 0 0 0

2 From this it can be concluded that col3(31) = 3 = 9 and col3(41) = 3. It follows that the figure-8 knot is not three-colorable. When reducing the matrices modulo 5, the resulting matrices are

−2 0 1 1  1 0 2 2 −2 1 1  1 0 4 1 −2 0 1 0 1 1 3 1 −2 1 ∼ 0 1 4 and   ∼   .      1 1 −2 0  0 0 0 0 1 1 −2 0 0 0     0 1 1 −2 0 0 0 0

2 It follows that col5(31) = 5 = 5 and col5(41) = 5 = 25, so the trefoil knot is not five-colorable. In the particular case of ` = 2, for any link L we have that at any crossing either all strands have the same color, or the two understrands have one color and the overstrand has another color. It follows that a knot K can only be colored trivially when using two colors, thus col2(K) = 2 and dim2(K) = 1. For a link L with r ≥ 2 components, one component can be colored with one color r and another component with another color, thus col2(L) = 2 and dim2(L) = r.

Proposition 2.3. For knots K1, K2 and prime p, we have that colp(K1)colp(K2) = p colp(K1#K2).

12 Proof. In the left of figure 21, the composition K1#K2 of knots K1 and K2 is shown, where the picture abstracts away from the knots using squares, and only shows the arcs connecting the knots. 0 Let K1 be K1 with an arc taken away, such that it has two outgoing arcs with the ends in a fixed 0 0 place. K1 is not a knot, but a so-called 1-tangle. In the right of figure 21, K1 is shown with below it a trivial knot, such that they only cross in the plane at the two outgoing arcs. Now, a p-coloring 0 0 0 of K1 can be extended to a p-coloring K1 with below it the unknot, because K1 is equivalent to 0 K1 with the unknot below it, as the unknot is splittable from the rest.

d a

K1 K2 K1 c b

0 Figure 21: Knot sum K1#K2 (left) and K1: K1 with trivial knot

0 For a p-coloring of this combination, let a and b be the colors of the outgoing arcs of K1, c and d the colors of the two parts of the unknot between the outgoing arcs. Then, using the equations that have to hold for the crossings, we find that 2a ≡ c + d mod p and 2b ≡ c + d mod p. It follows that a ≡ b. So for all p-colorings of K1#K2, the two connecting arcs must have the same 1 color. Thus colp(K1#K2) = p colp(K1) colp(K2) The coefficient matrix corresponding to the system of equations for a knot diagram is also called the crossing matrix for that diagram. To be more precise, when the n crossings and n strands of a knot diagram are labeled with {c1, ..., cn} and {s1, ..., sn} respectively, the crossing matrix M is defined as  −2 if s is an overstrand at crossing c ,  j i Mij = 1 if sj is an understrand at crossing ci, 0 otherwise.

Note that a shift of the labels and crossings along the knot results in a circular shift of the columns of the crossing matrix. With the above definition of a crossing matrix, the crossing matrix of a knot sum can be constructed using the crossing matrices of its components.

Proposition 2.4. Let D1 and D2 be knot diagrams of knots K1 and K2 with n and m strands respectively. Then there exist n × n coloring matrix MD1 and m × m coloring matrix MD2 such that the coloring matrix of K1#K2 corresponds to

 M 0 O  D1 n−1,m 1 ∗ 0 0 ... 0 1  1  .  O M 0   m−1,n D2  0 ... 0 1 1 ∗2 0

 M 0  0 Di Here, MD is MDi with its last row omitted, and MDi = . i 1 ∗i 1

Proof. Give D1 and D2 a fixed orientation. Label the strands of D1 and D2 with {s1, ..., sn} and {sn+1, ..., sn+m} respectively by following the orientation. Label the crossings of D1 and D2 such that strand si ends at crossing ci. Use this labeling to construct MD1 and MD2 . Take the knot sum K1#K2 by connecting the knot diagrams over the strands sn and sn+m such that the orientations match, and label the connecting strands, like in figure 22.

13 cn-1 cn+m cn-1 cn+m D s D D D 1 n 2 1 sn 2

s sn+m n+m

cn cn+m-1 cn cn+m-1

Figure 22: Performing a knot composition

In this way, the relations between strands at crossings c1, ..., cn−1, cn+1, ..., cn+m−1 are pre- served. It also means that, where first sn and sn+m ended at crossings cn and cn+m with over- strands si and sj and understrands s1 and sn+1 respectively, now sn and sn+m end at crossings cn+m and cn respectively. Constructing the coloring matrix using the described labeling then results in the desired matrix.

2.6 Knot determinant Because the system of equations for a knot diagram has at least ` trivial solutions when solved modulo `, we know that the columns of the crossing matrix are linearly dependent. Therefore the determinant of a crossing matrix is always 0. The determinant of a knot det(K) is defined as the absolute value of the determinant of a (n − 1) × (n − 1) minor crossing matrix, obtained by removing a row and a column of the crossing matrix of knot K. This determinant is independent from the choice of row and column that are removed of the crossing matrix, the labeling of a knot diagram, and the chosen knot diagram [13]. Therefore, the determinant is a knot invariant. Proposition 2.5. For knot K and prime p, we have that K is nontrivially p-colorable if and only if p divides det(K). Proof. Let M be an n × n crossing matrix for K. The corresponding system of equations has nontrivial solutions if and only if the dimension of the solution space is larger than 1 (proposition 2.2), and thus the rank of M must be smaller than n − 1. That means that every minor crossing matrix of M has determinant 0. Because we are calculating modulo p, it follows that det(K) ≡ 0 mod p should hold if and only if the system has nontrivial solutions. Thus K is p-colorable if and only if p divides det(K). When the above proposition is combined with the fact that a knot K only has trivial 2-colorings, as stated earlier, it follows that 2 - det(K). So, for any knot K, det(K) is odd, and in particular, det(K) 6= 0. In 1999, Louis Kauffmann and Frank Harary stated the following conjecture, also known as the Kauffman-Harary conjecture [10]. Conjecture 2.6. Let D be a reduced, alternating knot diagram of the knot K with det(K) = p prime. Then every nontrivial p-coloring of D assigns different colors to different strands of the diagram.

The conjecture was proven by Thomas Mattman and Pablo Solis in 2009 [14].

2.7 Alexander polynomial Let D be an oriented diagram of the knot K with n crossings and n+2 regions. Label the crossings c1, ..., cn and regions r1, ..., rn+2 Create a n×(n+2) matrix A, where, when taking the orientation into account,

14  0 if rj is not adjacent to crossing ci,  1 if r is on the right of the knot before passing the overstrand at c ,  j i Aij = −1 if rj is on the right of the knot after passing the overstrand at ci,  t if rj is on the left of the knot after passing the overstrand at ci,  −t if rj is on the left of the knot before passing the overstrand at ci. When two columns of A are removed and the determinant of the resulting n × n matrix is taken, the result is a polynomial a(t). This polynomial will differ depending on the two chosen columns, but only by a factor ±ts for some integer s. When dividing a(t) by the largest possible power of t and possibly multiplying by -1, such that the term of the lowest degree is a positive constant. The resulting polynomial ∆K (t) is called the Alexander polynomial of K, and is an invariant for K [3]. The determinant of a knot K is given by |∆K (−1)| [17].

3 Coding theory

In communication systems, codes are used to establish a reliable and efficient channel for data n transmission. A linear [n, k, d] code C over a field of size q, Fq, is a linear subspace of Fq of dimension k [18]. That is, for every pair of elements x, y ∈ C and pair of scalars a, b ∈ F , we have that ax + by ∈ C. The parameter n is the code length, and d is the minimum distance of C. The minimum distance of C is the minimum Hamming distance between any two distinct elements C, n where the Hamming distance d(x, y) between two distinct elements x, y ∈ Fq , which is a metric n on Fq , is given by the number of coordinates on which the elements differ. The elements of a code are called codewords, and the size of a code is the number of codewords that it contains, which equals qk.

Proposition 3.1 (Singleton bound). For a code C over Fq of length n and with minimum distance d, we have that |C| ≤ qn−d+1.

n−d+1 Proof. Suppose |C| > q . By the pigeonhole principle, there must be two codewords c1 and c2 in C such that c1 6= c2, that have the same value on n−d+1 positions, and thus have Hamming distance smaller than d. Thus |C| ≤ qn−d+1.  As, for a linear code C, |C| = qk, it follows that k ≤ n − d + 1. In a communication system with coding, a message m will first be encoded, resulting in a n codeword c ∈ Fq , which is sent over the channel. The output of the channel will then be a received word y, which may contain errors. Then y will be decoded into a codeword ˆc and a message ˆm, where it is aimed at having c = ˆc and m = ˆm. From this follows that the mapping m 7→ c is one-to-one.

n n Proposition 3.2. When decoding a received word y ∈ Fq into a codeword ˆc ∈ C ⊆ Fq after d−1 transmitting codeword c, up to b 2 c errors in a word can be recovered correctly, when d is the minimum Hamming distance of the C.

d−1 Proof. Let d(c, y) ≤ 2 . Decode y by taking a codeword in C closest to y with respect to the Hamming distance as ˆc. Suppose that ˆc 6= c. By the way ˆc was chosen, we have that d − 1 d(ˆc, y) ≤ d(c, y) ≤ . 2 From the triangle inequality it then follows that

d ≤ d(c, ˆc) ≤ d(c, y) + d(ˆc, y) = d − 1 .  Thus c = ˆc.

15 The Hamming weight w(x) of a codeword x ∈ F n is the number of nonzero entries in x. Note that d(x, y) = w(y − x).

Proposition 3.3. Let C be a linear [n, k, d] code over Fq. Then the minimum distance of C is equal to d = min{w(x)|x ∈C, x 6= 0}. Proof. As C is linear, we find that if x, y ∈ C, then y−x ∈ C. We also have that d(x, y) = w(y−x). From this follows that d = min{d(x, y)|x, y ∈ C, x 6= y} = min{w(y − x)|x, y ∈ C, x 6= y} = min{w(x)|x ∈ C, x 6= 0}.

The generator matrix G of a linear [n, k, d] code C over Fq is a k × n matrix, of which the rows form a basis of the code. This basis of k elements exists because C is a k-dimensional linear n k subspace of Fq . Each codeword c ∈ C can then be written as c = mG, for a unique m ∈ Fq . A simplified matrix G0 of the generator matrix G can be obtained by deleting all zero columns and all columns that are a scalar multiple from another column from G. The code resulting from G0 is called the simplified code C0 of C. An example of a code is {(0, 0, 0), (1, 1, 1)} over {0, 1}, which is the [3, 1, 3] repetition code. The idea is that, to make the channel more reliable, the data is sent repetitively. At the receiving end of the channel, the received message is decoded under the assumption that no more than 1 error occurred in the three repeated bits while being transferred over the channel. The [3, 1, 3] repetition code is generated by the generator matrix G = 1 1 1
. A parity check matrix of a linear [n, k, d] code C is an r × n matrix H over Fq, such that for n > every c ∈ Fq , c ∈ C ⇐⇒ Hc = 0. Thus the code C is the null space of H, and with the rank-nullity theorem, it follows that that r = rank(H) = n − k if the rows of of H are linearly independent. For the [3, 1, 3] repetition code, the parity matrix is 1 0 −1 . 0 1 −1

When in the r × n parity check matrix of a linear code every column has a fixed weight wc and every row has a fixed weight wr, where wc and wr are low, it is called a (wc, wr)−doubly-regular low density parity check (LDPC) code. With LDPC codes, the threshold of the amount of noise on the channel can be set arbitrarily close to the capacity of the channel. which is the theoretical maximum rate at which information can be transmitted over the channel, while the probability of mutations of data can be as low as desired. Two linear codes C1 and C2 over Fq with length n and dimension k are equivalent if the codewords from C2 can be obtained from the codewords of C1 by a combination of the following operations: 1. Permuting the n coordinates of the codewords;

2. Multiplying the coordinates at a fixed position of the codewords by a nonzero element of Fq. A linear code C is called cyclic if every circular shift of a codeword in C is also a codeword. That is, (c1, c2, ..., cn) ∈ C =⇒ (c2, ..., cn, c1) ∈ C.

3.1 Applications Error correcting codes are used in communication over a noisy channel, where errors may occur when transmitting a message. Giving the option of correcting errors is done by adding redundancy to the sent message. An example is a compact disc with some light scratches: some data may be lost due to the scratching, but may be recovered because of the redundant data stored on the disc. Another example can be found in spacecraft: when a photograph is sent from a satellite to Earth, LDPC codes may be used to ensure this happens correctly [4].

16 4 Colorings and codes

If (c1, c2, c3) denotes a three-coloring of the strands x1, x2 and x3 of a trefoil knot as depicted in figure 20 with colors c1, c2 and c3 respectively, it needs to hold that       −2 1 1 c1 0 3  1 −2 1  c2 = 0 in F3 . 1 1 −2 c3 0

n In general, for an `-coloring c of a knot with n strands, it needs to hold that Mc = 0 in F` . This gives rise to the idea to regard the possible colorings of a certain projection of a knot as a linear code C. In general, if a knot diagram is taken with strands x1, x2, ..., xn, an `-coloring of these strands n can be denoted by a codeword (c1, c2, ..., cn) ∈ C ⊆ Z` . Then the parity-check matrix of the code is given by the crossing matrix of the knot diagram. The dimension k of the code is larger than or equal to 1, as trivial colorings correspond to codewords are generated by 1 1 1 ...
. A knot is nontrivially `-colorable if and only if a corresponding [n, k, d] code has k > 1, which means that d < n, as follows from the Singleton bound. Note that each row and column of the coloring matrix of a knot has weight 3, and thus the code corresponding to a knot is a (3,3) LDPC code.

4.1 Reidemeister moves One may wonder whether the equivalence class of the simplification by considering consecutive columns of the generator matrix of the code C corresponding to a regular projection of a knot K will remain unchanged under the Reidemeister moves. When making the Reidemeister I move as pictured in figure 19, the generator matrices are related as follows:  |   | |  ··· x1 ··· ⇐⇒ ··· x1 x2 ··· . | | |

Clearly, these are the same when simplified, as x2 = x1. For the Reidemeister III move however, we obtain

 | | | | | |   | | | | | |  ··· x1 x2 x3 x4 x5 2x1 − x2 ··· ⇐⇒ ··· x1 x2 x3 x4 x5 2x3 − x2 ··· , | | | | | | | | | | | | with x4 = 2x3 − 1 and x5 = 2x3 − 2x1 + x2. Unfortunately, performing the Reidemeister moves now not only corresponds to permuting the columns of the generator matrix and multiplying the columns with a scalar, but also adding up columns. Thus the equivalence class of the simplification of the code will not remain unchanged under the Reidemeister moves.

4.2 Kauffman-Harary conjecture Using the Kauffman-Harary conjecture, it follows that p-colorings of alternating knot diagrams for a knot with prime determinant p can be described by a specific type of linear codes. In the proof of the conjecture, Mattman and Solis also prove the following proposition. Proposition 4.1. Let D be a reduced, alternating knot diagram of the knot K with det(K) = p. Then given any two distinct nontrivial colorings c1 and c2, there are integers a and b such that c2 ≡ a · 1 + b · c1 mod p. Here 1 is the trivial coloring, a vector of all 1’s.

17 Proposition 4.2. Let D be a reduced, alternating knot diagram with n strands of the knot K with det(K) = p prime. Then the [n, 2, n − 1] code over Zp denotes the possible p-colorings of D. Proof. According to the proven Kauffman-Harary conjecture, every nontrivial p-coloring of D assigns different colors to different strands. This means that the code C denoting the possible p-colorings of D has minimum distance d ≥ n−1, as there can be no more than one strand colored with color 0. From proposition 2.5 follows that the diagram is nontrivially p-colorable, as p divides det(K) = p. Thus the dimension of C is at least 2. From proposition 4.1 then follows that the dimension of C is equal to 2. 0 Let c = (c1, ..., cn) be a nontrivial p-coloring of the diagram. Then c = c−min{ci|1 ≤ i ≤ n}·1 is a nontrivial p-coloring of the diagram with one strand colored with the color 0, and w(c0) = n−1. Thus the minimum distance of C is n − 1.

4.3 Coloring of braids

Given a set of ` colors, say the elements of Z`, one can also look at `-colorings for braids. Again, these can be denoted as codewords, part of a linear code. Proposition 4.3. Let C be the linear code denoting the possible `-colorings of a braid with k strings. Then dim(C) = k.

Proof. Let x1, ..., xk be the strands that leave the top bar of a braid with k strings and let xk+1, ..., xn be the remaining strands of that braid. For every crossing it has to hold that the sum of the colors of the understrands equals twice the color of the overstrand modulo `. It follows that for a braid with k strings, when the colors of the k strands that leave the top bar are chosen, the colors of all the other strands depend uniquely on these. If an `-coloring of the braid is given by (c1, ..., cn), where strand xi has color ci, we thus have that cj is a linear combination aj,1c1 + ... + aj,kck of c1, ..., ck for k < j ≤ n. Then the generator matrix G of the code denoting the possible `-colorings of the braid is given by  1 if 1 ≤ i = j ≤ k,  Gij = aj,i if k < j ≤ n, 0 otherwise.

Then G is of the form (Ik|B), where Ik is the identity matrix of size k. This is an k × n matrix of rank k, so the dimension of the code is k.

5 Future Research

For future work, some aspects already touched upon in this report can be researched. Apart from the Alexander polynomial, there exist more polynomials related to knots, like the Kauffman polynomial and Jones polynomial. For codes there exist the weight enumerator and extended weight enumerator, and for graph there is the Tutte polynomial as an invariant. By having a look at these polynomials, one may find more about whether and how they are related. As for colorings, apart from coloring the strands, one can also consider the coloring of the regions created between the strands. Further, as the codes corresponding to braids and knots may be cyclic codes or LDPC codes, examining properties specific to these kinds of codes in relation to braids and knots may be of interest. Apart from focusing on several types of codes, one can also look at specific kinds of knots. Although the crossing matrix did not give a code that was invariant under the Reidemeister moves, there may be particular classes of knots that may result in codes that are invariants. Of interest would be to look into the class of knots with reduced, alternating diagrams.

18 Lastly, approaching the field of physics, where braids and error correcting codes meet in quan- tum computers may be a way to bring the knowledge of multiple mathematical fields into the world of physics.

6 Conclusion

In this report, several aspects of knot theory were highlighted, like the colorability, number of colorings and determinant of a knot as its invariants. Further, some concepts from coding theory are introduced. Finally, it was noted that colorings of knots and braids can be regarded as codes, and the special cases of colorings of reduced, alternating knot diagrams of knots with prime determinant were proven to be corresponding to [n, 2, n − 1] codes. Although, in general, knots that are equivalent under the Reidemeister moves do not necessarily correspond to equivalent codes, this may be the case for specific categories of knots. Future research on the properties of codes applied to knots may indicate this.

19 7 References

[1] Colin C. Adams. The Knot Book, An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society, Providence, 2004. [2] J. W. Alexander. A Lemma on Systems of Knotted Curves. Proceedings of the National Academy of Sciences, 9(3):93–95, 1923. [3] J.W. Alexander. Topological Invariants of Knots and Links. Transactions of the American Mathematical Society, 30(2):275–306, 1928. [4] Kenneth Andrews, Dariush Divsalar, and Bruce Moision. ECCs for Next Generation Space- craft Telecommand. In CCSDS Spring Meetings, 2013. [5] Emil Artin. Theorie der Z¨opfe. Abhandlungen aus dem Mathematischen Seminar der Universit¨at Hamburg, 4(1):47–72, 1947. [6] Clifford W. Ashley. The Ashley Book of Knots. Faber and Faber Limited, London, Boston, 12 edition, 1993. [7] David Austin. Knot Quandaries Quelled by Quandles.

[8] Rex Butler, Aaron Cohen, Matt Dalton, Lars Louder, Ryan Rettberg, and Allen Whitt. Explorations into Knot Theory: Colorability. July 2001. [9] Graham P. Collins. Computing with Quantum Knots. Scientific American, 294(4):56–63, 2009. [10] Frank Harary and Louis H Kauffman. Knots and Graphs I Arc Graphs and Colorings. Advances in Applied Math., 22(3):312–337, 1999. [11] W.B. Raymond Lickorish. An Introduction to Knot Theory. Springer, New York, 1997. [12] Peter Linnell. The Relation of Chemistry to Knot Theory, 2002. [13] Charles Livingston. Knot Theory, volume 24. Mathematical Association of America, 1 edition, 1993. [14] Thomas W Mattman and Pablo Solis. A proof of the Kauffman–Harary conjecture. Algebr. Geom. Topol., 9(4):2027–2039, 2009. [15] J´ozefH. Przytycki. 3-Coloring and Other Elementary Invariants of Knots. Banach Center Publications, 42(1):275–295, 2017. [16] Kurt Reidemeister. Elementare begr¨undungder knotentheorie. Abh. aus dem Math. Sem. der Uni. Hamburg, 5(1):24–32, December 1927. [17] Dale Rolfsen. Knots and Links. AMS Chelsea Publishing, Providence, 2003.

[18] Ron M. Roth. Introduction to Coding Theory. Cambridge University Press, New York, 2006. [19] De Witt Sumners. Untangling DNA. The Mathematical Intelligencer, 12(3):71–80, 1990. [20] William Thomson. On Vortex Atoms. Proceedings of the Royal Society of Edinburgh, 6:94– 105, 1867.

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