Algebraic and combinatorial techniques of knot theory based on arc colouring
By
Ayodele Arubi

Advisor: Dr. Alexei Vernitski

A project submitted in partial fulfilment of the requirements for the Degree of Bachelor of Science with Honours in Mathematics

University of Essex
Colchester, Essex

April 2015

Contents Abstract 4 Dedication 4 Acknowledgments 4 Introduction 5 The history of Knot theory 5 Brief history of knot theory 5 Development of the Knot Theory in Physics 7 Counting knots 9 The modern knot theory 12 Defining a Knot 14 Wild Knot 15 Definition of a Knot 16 Differentiating Knots 18 Orientation 18 Knot arithmetic 19 Modular arithmetic 20 Equivalence relation 21 Additivity Property 22 Multplicitivy Property 23 Knot Invariants 24 Reidmeister moves 24 Fox colouring 26 Dehn Colouring 28 Alexander-Briggs colourings 30 Implications of knot colourings 31 Other Knot invariants 31 The linking number 31 Quandles and Racks 35 Brief history about the Rack and Quandle invariant 35 Kei 36 Quandle 37 Racks 38 Definition 38 Examples of racks 39 Important definition of Racks 42 The free product. 42 The Cartesian product. 42 The disjoint union 42 Orbits and stabilizers of racks. 43 The operator group 43 The associated group 43 Bibliography 46

Abstract
This project narratively examines the history of the knot theory, its invariants and the theory of arc-colouring. This project will begin by providing a chronological history of the knot theory. It will show how the theory of knots originated from attempts to apply the study of knots to other fields especially Physics, for example, the structure of an atom and how it has emerged as a prominent field in mathematics over time.
We will then focus on the knot invariants; we still start however by introducing the modular arithmetic. We would define the Reidemeister moves after. The Reidemeister moves are essential in this project because, we’ll use the concept to show examples of arc-colouring like the Fox colouring were indeed knot invariants.
We will conclude by introducing the theory of Racks and quandle, giving a brief history of the concept and finish by providing basic definitions and examples.
Dedication
This project is dedicated to my father, Dr Jude Arubi. I would also like to dedicate this project to my family and friends for sticking by me and encouraging me during the process of writing this project.
Acknowledgments
Never in a millions years would I have imagined myself completing a dissertation. The only person that made this possible was my advisor, Dr Alexei Verntiski, who provided me with constant support and feedback on my project. He also provided useful and adequate sources needed to complete this project.
I would also like to thank the Department of Mathematical Science at the University of Essex for providing me with such an opportunity.

Introduction
The history of Knot theory
Brief history of knot theory
Knots have vied a fundamental part of our human history dating back to the prehistory era. Throughout history knots have vie its own part from the construction of iconic structures such as the Egyptian Pyramids to the Colosseum in Rome among others. Knots have continuously interested people for their aesthetics and religious symbolism. Furthermore, Knots appear in numerous Chinese artworks dating back several centuries BC. Additionally, they appear in Buddhism and numerous other religions and cultures.
It is very difficult to trace back when knots were originally discovered or were first encountered, however, the early records of such discoveries of knots can be accredited to the Greek physician Heraklas of the first century A.D.
He wrote books on surgeons’ knots and slings explaining ways to tie sixteen knots and nooses for surgical and orthopaedic purposes. His analysis on these knots could be perceived as the first attempt to understand knots and hence the knot theory. Although these knots were not mathematical knots, it could maybe make one think of the knot theory as an ancient concept, however, in regards to mathematics, knot theory is a fairly new concept.
Knot theory is a branch of topology. Topology was a concept developed by Gottfried Willhelm Leibniz in 1679. The German philosopher and mathematician laid the foundation for this new concept that he referred to as geometria situs (geometry of position). (Leibniz, 1850)
Definition: Topology is the mathematical study of the properties through deformations, twisting and stretching of objects. (Bugg & Ealick, 1990)
Leonhard Euler, provided the first ever example of Leibniz’s incipient mathematical idea. In his paper published in 1741, entitled “Solutio problematis ad geometriam situs pertinentis” (Hopkins & Wilson, 2004). Euler solving the bridges of Konigsberg problems realised that he required to not worry about the precise position of the bridges, instead figuring that the key information was to understand which properties derive from their reciprocal position. (Patone, 2011). He realised that magnitudes were not taken into considerations, nor does it involve calculation with quantities. (Euler, 1736)
In 1771, Alexander- Theophile Vandermonde wrote the first mathematical paper that introduced the idea of mathematical knots. He studied braids and knots as subject of the geometry of position in his paper entitled: “Remarques sur les problemes de ` situation” (Remarks on problems of positions). (Vandermonde, 1776) (Przytycki J. H., 1991)
Carl Friedrich Gauss, the German mathematician studies were vital in the origination of the mathematical knot theory. Between the years 1825-1844, Gauss focused on finding the classification of closed curves that had finite number of transverse self-intersections, that he stated as Tractfiguren (we could think of this idea like the knot projections).
His technique was to provide an orientation to the curves and then labelling the subsequent crossings with letter. As a result, a sequence was created that started by choosing a point on the Tractfiguren as the starting point. Therefore a curve with n crossing is said to have a sequence of length2n. The trefoil knot would have the sequence ABC ABC using this idea for example.
Figure 1: A Tracfiguren with the crossing sequence ABC ABC (Patone, 2011)
Figure 1: A Tracfiguren with the crossing sequence ABC ABC (Patone, 2011)

The sequence he created was only admissible for Tractfiguren with at most four crossing, as he soon discovered that his rules did not hold for Tractfiguren with five or more crossings which ended up being a huge limitation.
Gauss’ student Johann Benedikt Listing took an interest in knots. Listing’s published a monograph in 1847 entitled “Vorstudien zur topologie” (Listing, 1848) , within which he initial originated the term topology, with majority of the monograph being devoted to the investigation of knots and its classifications. He wanted to determine whether two knots projections showed the same knots using algebraic calculus.
The chirality of knots was an area of interest to him. A significant result in his discovery was that he became the first mathematician to express that the left hand trefoil and the right hand trefoil were not isotopic. Furthermore, he discovered that the figure eight knot and its mirror image were equivalent or isotopic.
Development of the Knot Theory in Physics
The first work on knot theory outside of Germany began in Scotland by physicist William Thomson (referred to as Lord Kelvin later) during the late 1860s. At the time, they were huge debates regarding the structure of an atom. With one side believing matter was composed by atoms while the others believed that matter was acting as waves.
In 1867, Thomson in a presentation, proposed that atoms were knotted vortices. Thomson’s paper was inspired by Herman von Helmholtz’s work (H.Helmholtz, 1858) on vortex motion and a demonstration by physicist Peter Tait exhibiting the properties of vortices using smoke rings.
He understood that the actual form of a vortex was not as vital as the hidden topological structure, and accepted a comprehension of such vortices would prompt a complete comprehension of matter (Patone, 2011).In other words, he was seeking ways to combining the two different general approaches to the structure of the atom. However, despite Tait’s crucial role in Thompson’s development of this idea, Tait was of the opinion that Thomson was heading in the wrong direction in trying to apply vortex motion in developing an atomic theory. Tait on the other hand believed that vortex motion’s primary application would be in the electromagnetism theory. However, despite Tait’s reluctance to agree with Thomson, he went ahead with the idea of an atom as vortices, this sparked an interest from mathematical physicist James Clerk Maxwell.
The physicist had been doing working in field of electricity and magnetism (electromagnetism) for a long time and was keen on how knot theory could be relevant to his field. He was open to the idea atoms were knotted vortices. Subsequently, he began writing to Thomson and Tait discussing his discoveries and ideas. They developed a keen interest in his ideas, as a matter of fact, he was the first scientist to notice that the simplest knot which comprised of a single strand was the trefoil and derived equations for the this curve. As a result, he went ahead to perceive a parameter in his equation that could tell if the trefoil produced was either left-handed or right-handed with respect to orientation. Furthermore, he asserted that it was not possible to change a left trefoil into a right trefoil one or the other way around.
In 1868, Maxwell began undertaking a genuine enthusiasm of the study topology. For a 3-dimensional space, Maxwell wanted to know when two knots projections denoted the same knot. To answer this inquiry, he formulated a labelling scheme for the knot projections’ crossing points. Afterward, Maxwell demonstrated that each knot projection contains a region where it is bounded by arcs that were less than four.
Note. An arc is the part of a knot from an undercrossing to the next undercrossing.
He started attempting to find all such regions. In the first case, where the region is bounded by a single arc, it was simply a turn or a twist required, which could be attained without altering the knot. For the second case, where two arcs bound the region, he discovered two possible outcomes. The two possible outcome were as follows, one region produced as a strand passed over another strand at two successive points or a region created as a strand passed over and then under another (Patone, 2011).This is illustrated in figure 2.
Figure 2: The regions bounded by less than four arcs (Patone, 2011)
Figure 2: The regions bounded by less than four arcs (Patone, 2011)

Lastly, the third case, with the region bounded by three arcs, there are also two possible cases (similar to the second case). This is also shown in the figure 2.
In Maxwell’s paper, he analysed a region bounded by three arcs. “In the first case, any one curve can be moved past the intersection of the other two without disturbing them. In the second case this 22 History of Knot Theory cannot be done and the intersection of two curves is a bar to the motion of the third in that direction.” (Epple M. , 1998). He also considered regions bounded by four arcs or more arcs, but was unable to make any real progress. Despite his discovery, it was evident that his knowledge was limited; for example, he claimed that any region whose boundary was partially right-handed and partially left-handed could be reduced in some way illustrating that he did not know of the existence of non-alternating knots.
Maxwell’s discovery was a huge step in the advancement of the knot theory. It is especially remarkable because Maxwell had unknowingly defined Reidemeister moves with his claims. The Reidemeister move was “discovered” nearly sixty year later by German mathematician Kurt Reidemeister. In other words, the Reidemeister move could easily be called the Maxwell moves as they are the same thing.
Counting knots As we stated earlier, Tait was initially reluctance to accept Thomson’s vortex theory of atoms despite Tait’s important role in Thomson’s idea. By the year 1876, Tait had created a knot table that include up to seven crossings. In any case, he comprehended that the difficulties of some the knots being generated would prevent them from being stable enough as vortices to represent atoms, that meant a tabulation of knots consisting of higher crossings was required (Tait P. , 1900).
Following in the footsteps of Maxwell and Gauss, Tait’s study of knots concentrated on developing ways to symbolically encode a knot projection crossing. As a result, Tait established an encoding scheme which resembled Gauss’ coding as opposed to Maxwell’s despite their affiliation. His scheme consisted of three principles, known as Tait conjectures. He used alternating knots to simplify this scheme.
Tait’s first conjecture stated that “An alternating diagram with no nugatory crossings, of an alternating knot realises the minimal number of crossings among all diagrams representing the link”. (Przytycki J. , 1998). Therefore, we can simply say that a reduced alternating knot diagram has minimal crossing number for the knot. He “proved” the above conjecture by showing that the removal of nugatory crossing was possible, and hence, it reduced the number of crossings as illustrated in the figure 3.
Figure 3: A picture of a nugatory
Figure 3: A picture of a nugatory

Note. Nugatory can be defined as a crossing that divided a diagram into two non-intersecting parts as shown in the figure 3.
His second conjecture cryptically states “if the simplest + - + - + - then irreducible” (Epple M. , 1998).Nowadays, we can interpret this conjecture as meaning that an alternating knot diagram without nugatory crossings, hence, those that separate two nontrivial distinct portions of the knot, which cannot be manipulated to have fewer crossings. We can also describe the second conjecture using writhe. We can describe the writhe using the following. If one equips a knot projection with an orientation, then each crossing looks, if observed from an angle, locally like it contains a positive crossing or negative crossing usually denoted by +1 and -1 respectively.
Figure 4: The linking numbers
Figure 4: The linking numbers

The two non-trivial knots- the trefoil and figure-eight cannot be projected with less crossings, as both are illustrated as alternating knots without nugatory crossing. However, despite Tait’s belief in the above, no prove for this existed until it was rigorously proved by Murasugi. (Murasugi, 1996)
Tait’s third conjecture was one he was not sure about at the time. The conjecture is referred to as Tait’s flyping conjecture that is normally stated “any two reduced alternating diagrams of a given knot are related via a sequence of flypes”,moves such as ones illustrated in Figure 5. The above flyping conjecture was proved by Menasco and Thistlehwaite in 1993 (Menasco & Thistlethwaite, 1991)
Figure 5: A flyping move (Patone, 2011)
Figure 5: A flyping move (Patone, 2011)

Following a pseudo-advertisement for someone to assist him with knot tabulation, Tait received a reply from Reverend Thomas Penyngton Kirkman, Rector of Croft, and Lancashire. Kirkman viewed the knot tabulation problem with regards to alternating knots only and hence formulated a table for these alternating knots with up to eleven crossing.
Due to Kirkman’s lack of interest in topology, Tait was required to pick up the slack of work to complete the table as Kirkman did not solve the problem of determining which alternating projections in Kirkman’s paper represented the same knots.
A former mathematics Ph.D. student Charles Newton Little (H.A. Newton at Yale University) sent a table to Tait containing 43 distinct non-alternating knots with up to 10 crossings in 1899. This table took him six years to complete.
For the next few years, many scientists continued to tabulate knots with various numbers of crossings, despite Tait’s retirement for knot tabulation. Mary Gertrude Hasmean partially extended this knot table in her doctoral dissertation at Bryn Mawr College in 1917. Following the discovery of some topological invariants, by the 1960s John H. Conway enumerated knots containing knots with 11 crossings. In 1983, C.H. Dowker and M. B. Thistlethwaite enumerated knots up to 13 crossings using computers. Nowadays, with the use of computers, knots enumerations have been made significantly easier prompting a fast development in the quantity of knots. For example, S. Rankin, J. Schermann, and O. Smith were able to discover an astonishing figure of 6,217,553,258 alternating knots with up to 22 crossings using a tabulation of all primes in July 2003. This is just one viewpoint in the rich history of knot theory; other various disciplines with respect to the knot theory have been developed over the past hundred years.
The modern knot theory
Thomson’s vortex atoms as we know was eventually abandoned, however, this did not stop mathematicians’ interest in the knot theory. There has been far too much advancement in the knot theory to be listed in this project. However, we will attempt to discuss as much of these major advancement as possible in this project.
Poincare introduced the Fundamental group in 1900. It was a noteworthy propel in the study of topology, as it made a route for the more-established tools of abstract algebra to be used by those examining the field of topology. Shortly after it was discovered, it was integrated into the knot theory. In 1908, Heinrich Tietze used the fundamental group of the complement or exterior of a knot in a 3-dimensional space, called the “knot group” in order to distinguish the unknot from the trefoil knot . ( (Tietze, 1908) translated by Biggs, Lloyd and Wilson in their book called “Graph theory” (Biggs, Lloyd, & Wilson, 1976)).
Austrian Wilhelm Wirtinger, the mathematician outlined a method for finding a knot group presentation in his lecture delivered at a meeting for the German Mathematical society in 1905, the method is referred to as the Wirtinger presentation (Przytycki J. H., 1991)
The German mathematician Max Dehn also developed an interest in the knot theory as he tried to prove Poincare conjecture. Unfortunately, he failed in his attempts to prove the Poincare conjecture, but he developed a different presentation to Wirtinger’s for creating the fundamental group of the exterior of a knot. He was however able to demonstration that a knot using the fundamental group, is nontrivial if and only if it is non- abelian.
Note. Non-abelian in mathematics is a group in which at least two element x and y of a group G such that x and y do not commute. In other words, mathematically it can be expressed: x*y≠y*x. Furthermore, he showed that a trefoil knot and its subsequent mirror image were distinct topologically confirming Maxwell’s theorem. Due to the World War I, any significant work was not carried out and it was not until the war ended that any important development in continued. In the 1920s, the Breakthrough in knot theory was due to two mathematicians James W. Alexander and Kurt Reidemeister at Princeton and Vienna respectively. They were studying the properties of the knot when they independently reached an identical knot invariant using complete different approaches with Alexander using homology groups and Reidemeister using fundamental groups.
Reidemeister demonstrated that all the projections of a knot were connected by a straightforward grouping of the three moves known as the Reidemeister moves shown in the figure 6.

Figure 6: The Reidemeister moves (MIT)
The outbreak of the Second World War halted the progression of the knot theory again. In April 1933, Reidemeister lost his professorship in Konigsberg for being “politically unreliable” (Epple Moritz, 1999). Ralph H. Fox was instrumental in the history of the Knot Theory following the conclusion of the war. The American succeeded in reshaping the way we viewed knot, providing greater access to the tools of topology for those studying knots. Hence, various new geometric knots were produced. Most notably, his own Fox colouring.
We have seen how the knot theory has progressed during its long history so far. We have also spoken about how new knot invariants have been formed using various mathematical approaches. In this next chapter, we shall formally define a knot. We will study different invariants, starting from fox colourings to the rack (quandle) invariants.

Defining a Knot
When you refer to the word “knot” to a non-mathematician they’ll probably refer to the shoelaces that helps prevent your shoes from falling off your feet or a troublesome tangle in a piece of string. If you were to ask a meteorologist, they’ll refer it as a unit of speed equal to one nautical mile. However, in mathematics, if we take two loose ends of a knot and attached them together, then we can such a knot a mathematical knot. In other words, a mathematical knot is different because the ends are joined together so it cannot be unfastened.
Although, this concept of a mathematical knot is easy to grasp using the above analogy, we cannot formally define a knot as ‘the loose ends attached together’ because it contains various discrepancy, for example, the size of the knot or the thickness of the knot etc. Hence, we’ll struggle to introduce any useful mathematical concepts from such a definition.
To formally define what a knot is, we’ll have to introduce some abstraction. Firstly, we need to think about the space in which knots exists. It is obvious that it does not exist on a straight line since we had early defined it as “the loose ends attached together”. The only space where a mathematical knot can exists is in a three-dimensional space denoted by R3 this is because of the places where the knot goes over or under itself therefore we cannot work with knots that lies in the Cartesian plane. Hence, leaving R3 as the only possible space where a knot can exist.
Secondly, another abstraction needed in defining what a knot is thickness. As mathematicians, we know that lines have no thickness therefore we can use that idea to say that knot has no thickness. We have to be cautious with this abstraction because it could lead to what is known as a wild knot. (Adams, 1994)
Wild Knot
Let us define a knot as a continuous function: f:[0,1]→R3 Such thatf0=f(1). In other words, a closed boundary and if fx=f(y) infers that: * x=y; or * x=1, y=0; or * x=0, y=1.
The above definition helps define knots quite accurately; however, it would allow us to have a knot with infinitely many twists, such knots are called wild knots. (Adams, 1994). Figure 7 shows a picture of the typical wild knot.

Figure 7: A typical wild knot (Adams, 1994)
Figure 7: A typical wild knot (Adams, 1994)

If f is differentiable everywhere then we could be able to remove such wild knots. Close to a point where the knots clustered up known as the wild point, the tangent vector changes rapidly, there is no continuous definition of the tangent as a result. Although, this requirement is adequate enough it can be tricky in other cases.
Definition of a Knot
We can use the idea of a closed polygonal curve to define a knot. Using Keller’s thesis (Keller reference) we can easily explain this concept. Suppose p and q are two distinct points in R3 . In other words, f:[p,q]→R3 Such that fp=f(q) (a closed boundary).
Suppose we have an ordered list of distinct points (p1, p2,p3,. . . , pn ) fromR3. We call the union of these distinct points of the line a closed polygonal curve because if we started from p1 for example, then we will end up with p1 by including the union[pn,p1].

Figure 8: Two polygonal curves
Figure 8: Two polygonal curves

The figure above shows two closed polygonal curves generated by the same set of four points but in various orders. This shows that a single set of points can define multiple closed polygonal curves. The above abstraction could lead us to define a knot as a closed polygonal curve; however, this would be fairly incorrect because for example, using Figure 8 we have some line segments intersecting at point which are not part of the four distinct points of the curve. This will contradict the idea of our mathematical knot because the knot cannot pass through itself.
To formally define what a mathematical knot is, we will have to introduce the idea of a simple closed polygonal curve. A simple closed polygonal curve does not curve itself. Therefore using this idea we can formally define a knot.
Definition A knot is a simple closed polygonal curve inR3. (Livingston, 1993)
The above definition of a knot only applies to tame knots. Tame knots are knots that are not considered to be wild knots. In order words, tame knots are knots that can be represented as polygons inR3.
Two knots are equivalent to one another if we can reproduce one of the knot from the other knot using series of ambient isotopy. Ambient isotopy refers to deformation of a knot by stretching or shrinking, moving and twisting; but not cutting or passing through itself to produce an equivalent knot.
Figure of an example of an equivalent knot
The simplest knot is called the unknot or the trivial knot. It contains 0 crossing and more complex knot can be transformed into this unknot. Figure 9 shows the trivial knot and two nontrivial knots called the trefoil (the simplest non-trivial knot with 3 crossing) and figure eight knots (4 crossing).
Figure 9: The unknot, the trefoil, and the figure-eight knot(MIT)
Figure 9: The unknot, the trefoil, and the figure-eight knot(MIT)

Knots can be placed into various classes. A very effective method of grouping these knots is using the number of crossings. The crossing number is accomplished using non-negative integer, which represents the minimum number of crossings that the knot can be denoted as having in a planar projection (Gay, 2006). The unknot is the only knot with 0-crossing and the simplest non-trivial knot has 3-crossings irrespective of its mirror images. (Livingston, 1993)
Amphicheiral knots are knots which are equivalent to its mirror imagine. A good example of such knot is the figure-eight knot while the trefoil is an example of a knot that is not. Furthermore, there are other classes of knots that exist but we are not going to concentrate on these various classes.
In addition, there exists other form of knots, like the braids (Brieskorn, 1988) or the links. The link is a collection of knots which do no intersect, but which may be linked together. In other words, knots are a special kind of links; therefore most of the theories presented will also apply to links.
Differentiating Knots We can use various knot invariant to show that two knots are equivalent, however, for the purpose of this section we’ll concentrate on distinguishing between two knots. Distinguishing between knots is a very difficult task, we can introduce the idea of orientation in other to make this task much easier.
Using the idea that your left hand and right hand are mirror images of each other. We can extend this idea to knots. In other words, we call a knot a chiral if it is not isotopic to its mirror image (Lafferty, 2013)
Orientation
Definition. The orientation on a knot is the direction that one moves around the knot. (Livingston, 1993)
An example of two trefoils having an orientation in different direction is pictured below. The trefoil on the left has a clockwise orientation while the right hand one has an anti-clockwise orientation. A knot is said to be oriented when it has been given an orientation. Furthermore, it should now be obvious that every knot can only possess one orientation.
Two oriented knots with opposite orientations assigned to the same knot may not be equivalent; however, this is a very difficult statement to proof. (Murasugi, 1996)
Knot arithmetic
We shall consider the addition of two knots to form a single knot.
Definition:
Let K1 and K2 be two oriented knots. The composition or connected sum of K1 and K2 denoted by K1#K2 is the oriented knot formed by attaching the knot K2 to K1 with respect with respect to the orientation of both knots. (Manturov, 2004)
In other words, we can achieve this composition by removing a small arc on each knot whilst still satisfying the following conditions: * Neither arc involves a crossing * The arcs are on the outside of the knots only
This will lead to the formation of two new arcs and these arcs do not form any new crossing. We could call this composition a proper composition as it satisfies all the conditions new. Below is a picture illustrating a composition of the knot K1 andK2. Here K1 is the figure-eight knot and K2 is the trefoil.
Figure 10: Composition of the figure-eight knot and the trefoil knot.( http://graphics.stanford.edu/courses/cs468-02-fall/projects/desanti.pdf)
Figure 10: Composition of the figure-eight knot and the trefoil knot.( http://graphics.stanford.edu/courses/cs468-02-fall/projects/desanti.pdf)

Proposition. Let K denote a knot, then the composition of K and the unkot equalK. We can express this as:
K#[unknot]=K
Proof. It is easy to proof the proposition above simply by letting the unknot acts as the integer 1, and hence, we can say the composition of a knot denoted by K and an unknot is equal to the knot.
Definition. A knot is said to be composite if it can be expressed as the composition of two knots.
Note. Neither of the knots could be the unknot.
Definition. A knot is said to be a prime knot if it cannot be expressed as the composition of two knots.
Modular arithmetic
An important concept when it comes to the Knot theory is Modular arithmetic. Carl Friedrich Gauss made further advancement with this concept during the 1800s in his book entitled: “Disquisitiones Arithmeticae” (Epple M. , 1995). He describes the modular Arithmetic as a concept in Mathematics where a system of arithmetic for integers “wraps around” upon reaching the modulus.
Using the 12-hour clock we easily define this concept. For example, using the concept of the 12-hour clock, in which a day (24 hours) is split into two 12- hour periods. If the time of the day is 5:00, then 15 hours later will be 8:00 as opposed to 20:00. This is because the clock (12-hour) has a modulus of 12 because once the clock reaches 12, it starts again. Furthermore, in this instance, 12 is congruent to 12, and it also congruent to 0. Suppose, the clock starts at 0:00 and 23 hours elapse, the time will be 11:00 (the same time as when we start at 12:00).
For any positive integer n, two integers x and y are said to be congruent modulo n, and can be expressed as x≡y(mod n) as, if the difference between the integer x and y have an integer multiple of n. The positive integer n is known as the modulus of the congruence where mod n is an equivalence relation. For instance 4≡14(mod 10) means that 2 is the remainder when 12 is divided by 10.
Definition. Let x, y ∈ Z and n ∈ N. We can say that “x is congruent to y modulo n”. Therefore, we can define the modulus of the congruence n as x≡y mod n if and only if n│(x-y).
In other words, x≡y(mod n) denotes that n divides(x-y). Furthermore, we can rewrite it as a=b+km for some integerk.
Note. The symbol │ denotes evenly divides.
So for instance, the above declaration that 4 and 14 are congruent modulo 10 is correct here by the explicit statement that10│14-4. A statement that can be shown further by the showing that14-4=10=10∙1.
Definition. Let x, y ∈ Z and n ∈ N. We can say that “x is not congruent to y modulo n”. Therefore, we can say that n is not the modulus of the congruence as x≢y mod n if and only if n∤(x-y).
Note. The symbol ∤ denotes not evenly divides.
Equivalence relation
The congruence modulo n has a relation which is an equivalence relation. Satisfying the following three propositions below:
Proposition 1. Let x ∈Z andn ∈ N. We can say that the relation is reflexive if x≡x mod n.
Proof. Using the identity that x-x=0 and n evenly divides 0 for any integer n. Thus, since we can rewrite x≡x (mod n)as n│x-x and hence completes the proof.
∎
Proposition 2. Let x, y ∈ Z and n ∈ N. We can say that the relation is symmetrical if x=y(mod n), then y=x(mod n).
Proof. . Let x,y,k∈Z and n ∈ N. We can rewrite x=y(mod n) as x-y=kn. Therefore, we can then rewrite y-x=-nk=n-k; So, by definition, y=x(mod n).
∎

Proposition 3. Let x, y,z ∈ Z and n ∈ N. We can say that relation is a transitivity if x≡y mod n And y≡z mod n
Then
x≡z mod n.
Proof. Let x, y,z ∈ Z and n,m,k ∈ N . Since we know that x≡y mod n can be rewritten as x-y=kn. Therefore, we can also rewrite y≡z mod n asy-z=nm. We can subtract the x≡y mod n from x≡z mod nto yield x-z=k-mn which means that x≡zmod n.
∎
Additivity Property
Proposition 4. Let w, x, y,z ∈ Z and n ∈ N. The additivity property of the modular congruence is such that if w≡y (mod n) and x≡z(mod n)
Then
w+x=y+zmod n.
Proof. Since we know that w≡y (mod n) can be rewritten as w-y=kn. Therefore, we can rewrite x≡zmod n as x-z=mn. Therefore, we have w-y+x-z=kn+mn=n(k+m) Hence, w+x=y+z(mod n)
∎
Multplicitivy Property
Proposition 5 Let w, x, y,z ∈ Z and n ∈ N. The multiplicitivity property of the modular congruence is such that if w≡y (mod n) and x≡z(mod n)
Then
wy≡xz mod n.
Proof. Since we know that -xy+xy=0. Therefore we have wy-xz=wy+0-xz =wy+-xy+xy-xz
=yw-x+x(y-z)
=ykn+bmn
=nyk+bm
And hence, we have n│(wy-xz) so wy≡xz mod n.
∎
The reason for studying the modular arithmetic concept is because it’s essential in understand various knot invariants we are going to explore in this project.
Knot Invariants
It is very difficult to show that two knots are the exact same. . Furthermore, we need an invariant in order to differentiate knots. A knot invariant is an entity (group, number, etc.) that can be associated to a knot and which is unchanged by any Reidemeister move. (Carter, Silver, & Williams, 2013)
Fox colourings of a knot provide the most straightforward but actual combinatorial invariants. In this chapter, we begin with by introducing the Fox colouring invariant followed by some definitions. The Fox colouring assigns colours to the 1-dimensional arcs of the knot (Carter, Silver, & Williams, 2013). We will also brief introduce what it’s said to be the sister colouring for the Fox colouring known as the Dehn colouring.
We’ll then introduce the Alexander-Briggs coloring . This colouring involves colouring the 0-dimensional crossing of the knot projection, with the regions of the projection determining the rules.
Firstly, in order to fully understand these invariants we must we start by defining the Reidemeister moves.
Reidmeister moves
In 1926, Kurt Reidemeister, devised a way for determining if two knots are equal. A concept independently demonstrated by J.W Alexander and G.B Briggs in 1927. The Reidemeister moves were three simple methods to deform knot projections by altering it’s number of crossings resulting in another knot that was equivalent to the original knot.
The three moves (alongside the ambient isotopy) required to deform a knot diagram to an enquivalent knot diagram of the same knot, are as follows: * Type I - allows us to twist the section of the knot to produce or remove a crossing. * Type II- allows us to poke part of the knot under or over another knot or unpoke a loop from under or over another to add or remove two crossings. * Type III- allows the sliding of a part of the knot from one side of a crossing to another side of a crossing.
Figure 11 shows all the three moves that makes up the Reidemiester moves.

Figure 11: The Reidemeister moves (MIT)
Definition. Two knots are said to be equivalent if you can repropduce the knot through a sequence of Reidmeister moves.
The eight figure knot is said to be equivalent to its mirror image. In other words, it is amphicheiral. Below is a diagram showing this using Reidmiester’s move.

Figure 12:The figure-eight and its mirror image

are equivalent (MIT)
The example above shows how important the Reidemeister moves are with regards to knot. It shows various manipulations of the Reidemeister moves can be used to show knots are equivalent etc.
We are going to move on to the knot colouring generated by these Reidemeister moves mention above.
Fox colouring
Before introducing the Fox colouring, we must first define the arc-coloring because the Fox coloring is an example of the arc-coloring.
Definition. The arc-coloringcan be defined as the assignment of colors 0, 1, 2, 3. . . ,n-1 to the arcs of K. (Carter, Silver, & Williams, 2013)
Note. K is the projection of the knot k, and n denotes the modulus.
Consider a classical diagram K and its arcs. We will colour each and every arc of this K with three colours; we call this colouring proper if at every classical crossing either all three arcs have the same colour or they all have three different colours. Using the figure 11 below of the trefoil, we can count the number of proper colourings (three in this case). Every classical knot diagram has three monochrome colourings.
Definition: A Fox colouring of a knot is that the assignment of integers to the arcs of the diagram such at every crossing double the whole number assigned to the over-arc equals the total of the integers assigned to the under-arcs meeting at this crossing. (Kauffman & Lopes, 2009)

Figure 13: The trefoil is tricolourable
Figure 13: The trefoil is tricolourable

We equivalently express this equality, as “the total of the under-arcs equals double the over-arc”. In other words, we can represent this expression by the following equation:
2b=a+c (mod n)
Note. a,b and c denotes the labels of the arc, with b being the overcrossing.
Definition: A Fox coloring is said to be trivial if each arc is labelled the same colour.
Theorem. The Fox coloringis a knot invariant
Proof. In order to prove this theorem, we must show that the Reidemeister moves remain unchanged with respect to the colouring. We will use the same style as Martina’s dissertation (Patone, 2011) to show this proof.
Firstly, suppose we start by introducing a crossing, so, looking at the first Reidemeister move , we can simply assign the same colour to every stand, and hence the new formed crossing produced will fulfil the prerequisites for the Fox coloring. Similarly, if we remove a crossing using the first Reidemeister move it will still remain a Fox coloring. Below is a figure 14 showing the first Reidemeister move preserving theFox coloring.

Figure 14: The first Reidemeister move preserves the Fox colouring (Patone, 2011)
Figure 14: The first Reidemeister move preserves the Fox colouring (Patone, 2011)

Secondly, examining Reidemeister’s second move, suppose we introduce two new crossings while the original two strands were two different colours, if we were to change the colour of the newly formed strand, it will form a third colour, hence, fulfilling the prerequisites for the Fox coloring. Furthermore, if the original strands possessed the same colour, we could keep the newly formed strand and the crossing formed the same colour as the original colour, hence, the Fox coloring prerequisites are satisfied.
In addition, the Fox colouring condition will be satisfied if we were to decrease the amount of original crossings by two. This is because either each one of the strands will be the same colour, therefore, we can colour each of these strands the same colour, or three different colours at every crossings formed, so we can colour the two resulting strands two distinct colours. Therefore, the Fox coloring is still preserved.

Figure 15: Reidemeister’s second move preserves the Fox colouring (Patone, 2011)
Figure 15: Reidemeister’s second move preserves the Fox colouring (Patone, 2011)

Lastly, we can confirm Reidemeister’s third move preserves the Fox coloringby five different ways. The figure 16 below shows all the five possible moves that shows the Reidemeister move preserves the Fox coloring.
Figure 16: Reidemeister’s third move preserves the Fox colouring (Patone, 2011)
Figure 16: Reidemeister’s third move preserves the Fox colouring (Patone, 2011)

Dehn Colouring
The Fox colouring has a twin sibling called the Dehn colouring. The difference between both colourings is that the Fox colouring assigns elements to the 1-dimensional arcs of a knot project; the Dehn colouring assigns elements to the 2-dimensional arcs of a knot projection.
The rule for the Dehn colouring is denoted by the equation: a+b=c+d (mod n) (Carter, Silver, & Williams, 2013)
Note. a,b,c and d denotes arcs labels of a projection.
The equation is generated by the Dehn relation. We can describe the Dehn relation as follows: if Ri,Rj,Rk,Rl are the regions at a crossing, then the associated relation according to Max Dehn isRiRj-1RkRl-1. (Carter, Silver, & Williams, 2013)
Figure 17: Dehn relation and Dehn n-colouring rule (Carter, Silver, & Williams, 2013)
Figure 17: Dehn relation and Dehn n-colouring rule (Carter, Silver, & Williams, 2013)

Lemma: An arc-coloringof K is said to be “conservative if and only if it is a Fox coloring.” (Carter, Silver, & Williams, 2013)
Proof For the purpose of this proof, suppose that the arc-coloring is aFox coloring. Then it is adequate and essential to show that integration along any closed path γ yields the original colour of that region.
Using induction on the number N of crossings enclosed by the path γ. Suppose N equates to zero, and then the result is obviously a constant. Considering a small circle denoted by δ around a crossing that is encircled by the path γ. The addition of these two variables (γ andδ) yields another closed path denoted byγ, Therefore, integrating γ' and γ yields matching results, however, γ' encloses N-1 crossings.
Conversely, suppose the arc-coloring is not aFox coloring. Then, the arc-coloring is said to be not conservative because if we integrate along a closed path denoted by γ for example, it will not yield the initial colour. ∎
The above lemma is crucial as it gives a well-defined method for passing from a Fox n-colouring to a Dehn colouring.
Alexander-Briggs colourings This colouring was invented by J.W. Alexander and G.B Briggs (Alexander & Briggs, 1926-27). They showed an oriented knot by its basic projection in the planes as a 4-valent graph, marking corners with minor points. For that reason, we shall refer to the diagrams regarding the Alexander-Briggs colouring as the knots’ Tait diagram .
Let K be a Tait diagram. We define a vertex-colouring as the assignment of colours {0,1, . . ., n-1} (mod n) to the vertices of K. A vertex colouring is said to be an Alexander-Briggs colouring if in each region the sum of the colours of the un-dotted vertices subtracted from the colours of the dotted vertices- the weighted vertex sum-vanishes. (Carter, Silver, & Williams, 2013). Figure 18 gives an example of a typical Alexander-Briggs and also shows the Tait’s dot-notation.

Figure 18: A typical Alexander Briggs colouring (coupled with Tait's notation) (Carter, Silver, & Williams, 2013)
Figure 18: A typical Alexander Briggs colouring (coupled with Tait's notation) (Carter, Silver, & Williams, 2013)

Implications of knot colourings
The methods of knot colouring described above are only a minor aspect of knots. There are various directions that can be generated from these colourings.
Fox colourings are a special type of quandle colourings. We are to define a quandle later on in the project. Winker in his dissertation was the brains behind the idea of using this quandle colouring to identify knotting (Winker, 1984) and subsequently discussed by Kauffman and Harary (Harary & Kauffman, 1999).
Furthermore, the Fox, Dehn and Alexander Briggs colourings are very crucial to the Knot theory because they help distinguish between knots albeit with various limitations.
Other Knot invariants
They are various other knot invariants that have been discovered over the years. In this section, some of these invariants will be defined very briefly.
The linking number
A link is a set of looped knots all tangled up together (Patone, 2011). Below are projections of two important links, known as the “Whitehead link and the Borromean rings”.
Figure 19: The Whitehead link and the Borromean rings (Patone, 2011)
Figure 19: The Whitehead link and the Borromean rings (Patone, 2011)

The Whitehead link composes of two loops knotted together; therefore we say that it is a link of two components. The Borromean rings on the other hand composes of three loops knotted together, therefore we can call it a link consisting of three components. A knot can be described as a single component of a link.
The equivalence of two knots can be easily checked. This is performed by counting the components of the link. Suppose, the numbers of components differs, the two links are different. However, we cannot say that if the number of component were the same, the two links are the same. This is because for example, considering two types of links: Hopf link and the unlink in figure 18, although these two links contain two components each, they cannot be equivalent. The unlink cannot be split and hence cannot be transformed into the Hopf link.
Figure 20:The two component unlink and the Hopf link
Figure 20:The two component unlink and the Hopf link

Therefore, in other to check that two links are equivalent, we have to introduce the invariant known as the linking number. There are two possible configurations at each crossing point of an oriented link. At a crossing point denoted by, we denote the sign(c) +1 or sign(c) -1 where +1 represent a positive crossing and -1 denote a negative crossing.
Hence, we can define a linking number as follows:
Definition Let D denote the diagram of a two component link, denoted by K={K1,K2}. Then, the crossing points of D at which the projections of K1 and K2 intersect are denoted by c1c2, . . . cm. Then we have:
12{signc1+signc2+. . . +signcm}
This linking number D, is denoted by the notationlk(D).
Note. If K1 and K2 have the same intersections at a crossing point we assign the value of 0 to it.
The linking number has been has been listed as an invariant in this project, but how do we know this linking number is indeed an invariant? Let’s check:
Theorem. The linking number denoted by lk(D) is a knot invariant.
Proof. In order to show the linking number is an invariant it is essential to show that it remains unchanged with respect to the Reidemeister moves.
Type I. The Reidemeister move does not affect the linking number because the crossing that is removed or added is within the arc-itself. In other words, one side will have an extra crossing but since it’s self-crossing then the linking number is not affected.
Type II. If the two strands correspond to same components, then the move has no effect on the linking number. Now, suppose they had different components with one of the crossings contributing to +1 and the other -1 on the other one. The total contribution would be 0. Suppose, we altered the orientation of one of the strands, we would still be left with +1 and -1 which has a total contribution of 0. Therefore the type II Reidemeister move leave the linking number unchanged.
Type III. If the three strands correspond to the same components, then the move has no effect on the linking number. Now, suppose the strand were given an orientation with +1 and -1 being assigned to all the crossing, it is obvious that moving the strand over with respect to the Reidemeister’s type III condition does not affect the number of +1 or -1, and so we can conclude that the linking number holds.
So, we can conclude that the linking number is indeed an invariant to an extent. However, it is not a perfect invariant because it does not help separate two different links with same linking number. For example, the unlink of two components has a link of zero and the Whitehead link also has a link of zero so according to the linking number, these knots are equivalent which is incorrect.
The linking number is an example of a combinatorial knot invariant that includes the likes of the Bridge number, the Writhe Number, the Unknotting number and the Crossing number. All of the invariants can all be verified in identical manner.

Quandles and Racks
A brief history
In 1942, Takasaki presented a notion called kei, eventually become the involutory quandle. They are both algebraic entities that consists of the non-empty set and a binary operation on it operating from the right translation. (Kamada, 2001)
The notion of the quandle is generated when we drop the above condition of the kei, a concept introduced by Joyce in 1982. His journal was entitled “A classifying invariant of knots: the knot quandle” (Joyce, 1982). He linked his concept of a quandle to the knot theory, forming what is now known as the knot quandle and using the Reidemeister move, he ahead to prove that the knot quandle was a complete invariant. (Kamada, 2001). In addition, independently, Matveev proved some similar result around the same time. A concept he referred to as the distributive grouppoid (however, for the purpose of this project we shall refer to the invariant as the quandle) (Matveev, 1984). In 1988, Brieskorn introduced the notion of an automorphic set. (Brieskorn, 1988), Fenn and Rourke then introduced the notion of a rack. (Fenn & Rourke, 1992). Racks and automorphic sets are the same concept, with the only difference being that they operated on the right translations and the left translations, respectively. Anyway, this notiond was made possible by removing the first condition of a quandle (for every a M we have a৹a=a).
Racks and quandle are solid and useful invariants, however, it is quite hard to distinguish using direct calculations.
The simplest way of proving, for example, non-triviality of the trefoil knot is the usage of the so-called colouring invariant. In the previous chapter, we introduced some types of colouring invariants like the Fox colouring, Dehn colouring etc. An important definition in regards to colouring invariants is the proper colouring. (Manturov, 2004)
Definition: Let D denote the diagram of an oriented Knot denoted by K, we say it is a proper colouring if there is a way of associating some colour with arc of D in such a way that each overcrossing arc denoted by b, undercrossing arc lying on the left hand denoted by a and undercrossing lying on the right hand denoted by c, the following relation holds: a৹b৹c This relation is shown in the figure below. The relation is called the rule of colouring of a knot.

Figure 21: The crossing rule of a knot (Manturov, 2004)
Figure 21: The crossing rule of a knot (Manturov, 2004)

The important question here, what conditions are necessary for the above definition to be an invariant?
In, this chapter, we will briefly introduction into quandles and hence a racks. We will use this concept to show that the proper colouring is indeed a invariant under the Reidemeister moves.
Note. The majority of the ideas presented in this chapter are from Fenn Roger and Rourke Colin’s paper on the rack entitled “Racks and links in codimension two” (Fenn & Rourke, 1992) . Any other references will be stated.
Kei
A kei, , is a non-empty set with a binary operation a,b↦a৹ b satisfying the following axioms:
(K1) for every a that is an element of M , we have a৹a=a;
(K2) for every a,b that are elements of M, the relation (a৹b)৹b=a holds.
(K3) for every a,b,c that are element of M the relationa৹b৹c=(a৹c)৹(b৹c) holds.
Quandle
We are going to define a quandle using Manturov’s book “knot theory” (Manturov, 2004). Hence, we can define a quandle (a distributive grouppoid) as a set M, endowed with a binary operation ◦, satisfying the following properties:
(Q1) Idempotency: for every a that is an element of M , we have a৹a=a;
(Q2) The existence of a left inverse: for every a,b that are elements of M, the relation (a৹b)৹b=a holds
(Q3) Right self-distributivity: for every a,b,c that are element of M the relationa৹b৹c=(a৹c)৹(b৹c) holds.
Proposition. The number of proper colourings by elements of any quandle is a link invariant. (Manturov, 2004)
Proof: The proof of this proposition is easily proved using the same ideology as we have described for the Fox colouring for example. Figure 22 illustrating this statement is pictured below.

Figure 22:The Reidemeister moves show a quandle is an invariant (Carter & Saito)
Figure 22:The Reidemeister moves show a quandle is an invariant (Carter & Saito)

Note. There are various forms of notion relating to the quandle. One notable alternative is the exponential notation where a৹b can be expressed as ab, and a∘-1b is written as ab. Furthermore, in figure 22 we replaced ৹ with * which is another form of notation for the quandle.
We are going to define what a rack is and include some examples of racks in the following section.
Racks
Definition
A rack, R, is a non-empty set with a binary operation a,b↦a৹ b satisfying the following conditions:
(R1) For any a,b that is an element of R there exists a unique c that is an element R such that a=c৹b
(R2) For any a,b,c that are elements of R, we have a৹b৹c=a৹c৹(b৹c)
Note. A quandle is a rack using the definition.
Definition There exists a rack R such that the mirror rack denoted by R has binary operationa ৹ b.
Definition Rack homomorphisms, Rack automorphisms and Subracks are defined in the natural way. (Ryder, 1993)

Proposition. The number of proper colourings by elements of any rack is a link invariant.
Proof. This is very similar to the proof of the quandle. The only difference is that it does not include the first condition of the quandle (from the definition). Figure 23 below proves the proposition above.

Figure 23:The Reidemeister moves show a rack is an invariant (Carter & Saito) Figure 23:The Reidemeister moves show a rack is an invariant (Carter & Saito)

Examples of racks
The conjugation rack.
Let M be a group. We can define a rack as the elements of G such that a∘b=bab. (Where b represents b-1).

Therefore we have a rack in terms of: a∘b∘c=cbabc and a∘c∘b∘c=(cac)∘(cbc) =cbccaccbc
=cbabc.
The rack defined above is called the conjugation rack and is written ConjM.
Example. Below is the conjugation rack Conj (S3).

| id | a | a2 | b | ab | a2b | id | id | id | id | id | id | id | a | a | a | a | a2 | a2 | a2 | a2 | a2 | a2 | a2 | a | a | a | b | b | ab | a2b | b | a2b | ab | ab | ab | a2b | b | a2b | ab | b | a2b | a2b | b | ab | ab | b | a2b |

Trivial rack
The trivial rack of order n, written Tn, is the rack with n elements, all of which are trivial as operators. In other words, the rack structure defined on {0,…,n-1} by allowing
(a∘b)∶=a.
Example. Below is the trivial rack of order four. | a | b | c | d | a | a | a | a | a | b | b | b | b | b | c | c | c | c | c | d | d | d | d | d |

The dihedral rack.
Any union of contumacy classes in a group, M, form a rack with conjugation as the operator (ConjM). In particular, the set of reflections denoted by Rn in the dihedral group, D2n, forms a subrack of ConjD2n. We refer to this particular rack as the dihedral rack of order n, written D2n..
Example. Below is the dihedral rack D3 | a | b | c | a | a | c | b | b | c | b | a | c | b | a | c |

Important definition of Racks
The free product.
Let G and H be two racks. The free product of these racks can be defined as an operation that takes two racks G and H and constructs a new rack G*H.
More precisely, G*H consists of elements of the form (g∘w) or (h∘w) where g is an element of G, h is an element of H and w is an element of As (G) *As (H). (Where As G and As H are the associated groups of G and H respectively).
The Cartesian product.
The Cartesian product of two racks, G and H, is defined by every ordered pair taking the form (g, h) where g is an element of G and h is an element of H such that: g,h∘g',h'=(g∙g',h*h') Where ∙ denotes the operation for G and * denotes the operation for H.
The disjoint union
The disjoint union of two racks, G and H is defined by Brieskorn (Brieskorn, 1988). It has the disjoint union of the sets Gs and Hs as elements with the action given by: a∘b=a∘b a,b∈R or a,b∈S. a∘b=a a∈R,b∈S or a ∈S. b∈R.

Orbits and stabilizers of racks.
For the examples below, we are going to use Ryder’s dissertation (Ryder, 1993).
Definition The orbit of an element a of a rack R, denoted orbRa, is the set of all elements b which are such that there exists an operator w wherea∘w=b. An unspecified rack R , can easily denote the notation as orba.
Definition A single orbit rack is called as a transitive rack.
Definition Let S denote a subset of the set of elements of a rack R. The orbit of S, written orbR(S), is the set of all elements b which are such that there exists an operator w and an element s in S with s∘w=b. If R is obvious or unspecified, we can easily denote the notation as orbS.
Definition The stabilizer of an element a of a rack R, denoted stabRa, is the set of all operators w which are such that a∘w=a. We can easily denote the notation stabS If R is said to be unspecified.
The operator group
The group of operators of a rack R, is called the operator group of R, denoted by OpR. (Ryder, 1993)

The associated group
The associated group, of a rack R, is defined as the operation of a rack R as conjugation. In other words, we can define the associated group by the group bab=c. Hence, using the intuition that any rack R can be presented in the form: a,b,c,… a∘b=c,…).
We can therefore present the associated group of the rack R by: a,b,c,… bab=c,…).
Note. Here b represents the inverse of b (b-1). Using Wirtinger’s presentation generates the above presentation of the rack R.
Proposition. Let R be a rack and let G be a group. The associated group of a rack R is a definite invariant of R.
The map F, from the category of racks to the category of groups, denoted by:
R→AsR,
Is a functor and is a left adjoint to the functor sending a group G to the conjugation rack denoted ConjG. there is a natural identification between the sets Hom (AsR, G) and Hom R, ConjG. Therefore, given any racks homomorphism f: R→Conj (G), there exists a unique group homomorphism f1: AsR→G which ensures the following diagram commute: R η AsR ↓f ↓ f1 ConjG id G
Where η is the natural map.
The above proposition is called the Universal Property of the Associated Group.
Proof Suppose R is any rack and Let G be any group. Then, we can let ϕ: F (R) →G be the homomorphism defined on the free group on R by f. Using the hypothesis: ϕ(a∘bbba)=1 For every a, b in R.
Therefore, we can derive that ϕ factor through a unique homomorphism f1: AsR→G of groups, which ensure the diagram, commutes. The universal property explains the uniqueness of the associated group.
Note. ϕ (a∘bbba)=1 can be derived by the basic properties of a rack.
We can define the associated group of a rack R using the quotient group of the free group on elements of R and where K is the normal subgroup of F (R) generated by the words a∘bbba wherea, b∈R. Therefore, AsR=F (R)/K. In addition, we can represent the operator group as follows: OpR=F (R)/N. In other words, the operator group is equal to the associated group divided by N, the subgroup of the free group of R, which acts trivially on R. As a result of the above derivation, we have the following sequences:
K→FR→AsR,
N→F(R)→Op(R) and NK→AsR→OpR.
.

Bibliography
Adams, C. C. (1994). The knot book. New York.
Alexander, J., & Briggs, G. (1926-27). On types of knotted curves. Annals of Math., 28, 562-586.
Biggs, N. L., Lloyd, E. K., & Wilson, R. J. (1976). Graph Theory. Oxford: Clarendon Press.
Biggs, N., Lloyd, E., & Wilson, R. (1976). Graph Theory 1736-1936. Oxford: Clarendon Press.
Brieskorn, E. (1988). Automorphic sets, braids and singularities. Contemporary maths, 78, 45-115.
Bugg, C. E., & Ealick, S. E. (1990). Crystallographic and modeling methods in molecular design. New York: Springer-Verlag.
Carter, J. S., & Saito, M. (n.d.). Coloring Knot Diagrams, Knotted Surfaces, and Quandles. Alabama: University of SOuth Alabama. Retrieved April 17, 2015, from http://www.southalabama.edu/mathstat/personal_pages/carter/Osaka.pdf
Carter, J. S., Silver, D. S., & Williams, S. G. (2013). Three dimensions of knot coloring.
Carter, J., Jelsovsky, D., Kamada, S., Langford, L., & Saito, M. (2003). Quandle cohomology and state-sum invariants of knotted curves and surfaces. Trans. Amer. Math. Soc, 3947-3989.
Epple Moritz. (1999). Geometric aspects in the development of knot theory.
Epple, M. (1995). Branch points of Algebraic Functions and the Beginnings of Modern Knot Theory. Historia Mathematica, 371-401.
Epple, M. (1998). Topology, matter and space 1.Topological notions in 19th-century natural philosophy. Archive of History of Exact Sciences, 52, 297-382.
Euler, L. (1736). Solutio problematis ad geometriam situs pertinentis. ommentarii Academiae Scientiarum Imperialis Petropolitanae, 8, 128-140.
Fenn, R., & Rourke, C. (1992). Racks and links in codimension two. Journal of Knot Theory and its Ramifications , 01(04), 343-406.
Gay, D. (2006). Explorations in Topology. Arizona.
H.Helmholtz. (1858). Uber integrale der hydrodynamischen gleichungen, welche den wirbelbewegungen entsprechen. Journal fur die reine und angewandte Mathematik, 25-55.
Harary, F., & Kauffman, L. (1999). Knots and graphs I- arc graphs and colorings. Advances in Applied Mathematics, 22, 312-337.
Hopkins, B., & Wilson, R. (2004). The Truth about Konigsberg. The College Mathematics Journal, 198-207.
Joyce, D. (1982). A classifying invariant of knots: the knot quandle. Journal of Pure and Applied Algebra, 37-65.
Kamada, S. (2001). Knots invariants dervied from quandles and racks. Geometry & Topology Monographs, 103-117.
Kauffman, L., & Lopes, P. (2009). Determinants of Rational Knots. Discrete Mathematics & Theoretical Computer Science.
Lafferty, K. (2013). The Three-Variable Bracket Polynomial for Reduced, Alternating Links .
Leibniz, G. (1850). Math. Schriften. 1 Abt., 2, 19-20.
Listing, J. (1848). Vorstudien zur topologie. Gottingen: Vandenhoeck und Ruprecht.
Livingston, C. (1993). Knot theory. Washington, DC: Mathematical Association of America.
Manturov, V. O. (2004). Knot theory. Boca Raton: Chapman & Hall/CRC.
Matveev, S. (1984). Distributive groupoids in knot theory (Russian). Mathematics of the USSr-Sbornik, 47(1), 73-88.
Menasco, W., & Thistlethwaite, M. (1991). The Tait flyping conjecture. Bulletin of the American Mathematical Society, 25(2), 403-413.
Murasugi, K. (1996). Knot theory and its applications. Boston: Birkhauser.
Patone, M. (2011). Knot Theory and its applications. Bologna: Univeristy of Bologna.
Przytycki, J. (1998). Classical roots of Knot Theory. Chaos, Solitons and Fractals, 531-545.
Przytycki, J. H. (1991). Knot theory from Vandermonde to Jones. National Congress of the Mexican Mathematical Society, 11, 173-185.
Ryder, H. (1993). The structure of Racks. Warwick: University of Warwick.
Tait, P. (1867). 485-511.
Tait, P. (1900). On knots I, II, III. Scientific papers, 1, 273-347.
Tietze, H. (1908). Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten. Monatsh. Math. Phys., 19(1), 1-118.
Vandermonde, A. (1776). Remarques sur les problemes de situation. (N. L. Biggs, E. Lloyds, & R. Wilson, Trans.) Histoire de l'academic Royale des Sciences (France), 566-74.
Winker, S. (1984). Quandles, knots invariants and the n-fold branched cover. Chicago: University of Illinois.

--------------------------------------------
[ 1 ]. Two knots are said to be isotopic if they are said to equivalent to each other.
[ 2 ]. Vortex is a mass of water or air that spins around very fast and pulls objects into its empty centre.
[ 3 ]. Alternating knots is a knot where the crossings alternate over, under, over and under as oen travels along each component of the link.
[ 4 ]. A group is an algebraic structure consisting of a set of elements together with an operation that combines any two element to create a third element.
[ 5 ]. Homomorphism functions are functions between two groups that “preserve” the group operations.
[ 6 ]. The free group of a set S is a set that consist of all terms that can be created from members of S, considering two expressions are different unless an element and its inverse.