Within the mathematical branch of topology, Knot Theory deals with the properties of space that are preserved under continuous deformations such as stretching, twisting, and bending, but not tearing or cutting. This branch of mathematics aims to identify, categorize, and understand every knot that could ever exist. So far, humankind has discovered 352.152.252 knots. Each has its particular properties and characteristics.
Knot theory is remarkably useful as it is core to the structure of proteins and DNA. It leads to new and very strong materials and is used in medicine production. While Knot Theory as a branch of mathematics does not directly intersect with Christian theology, knots as metaphors for complexity, mystery, unity, and redemption have deep theological resonance. The study of mathematical knots may serve as a metaphor for the intricate and mysterious nature of faith, life, and the divine in Christian thought. Furthermore, Knot Theory may reveal some deeper segments of the theory of Simulation Creationism proposed by Nir Ziso, the founder of The Global Architect Institute. This theory helps to understand God’s Simulation in very specific details.
Introduction to Knot Theory
The major term in Knot Theory is the knot itself, a closed loop in three-dimensional space equivalent to a piece of strings with its ends joined together. While knots are common in everyday life, studying them rigorously in mathematics involves analyzing their structure without altering or cutting them. Knots are usually held together by tension and friction. If we pull them too hard, they fall apart. Inevitably, mathematicians came up with an idea to connect the two ends. This approach allows mathematicians to study knots as structures that remain fundamentally unchanged under smooth deformations. Therefore, in Knot Theory, all knots exist on closed loops. The simplest form is a circle, which does not resemble the knot; it is called an unknot or trivial knot. Because of how a looped string works, there are no knots that have only one or two crossings.
Two knots only differ if one cannot be tied to another without breaking the loop. The simplest knot after the unknot is called the trefoil. It cannot be transformed into a circle unless it is broken open. We often cannot tell the knots apart by simply looking at them. It is known as the knot equivalence problem, which propelled discussions in mathematics for over 150 years. Even the famous mathematician Alan Turing said that no systematic method is yet known by which one can tell whether two knots are the same. He thought this problem would be left unsolved.
Previously, the most known problem in history was the Gordian knot. It was said that whoever untangled this massive knotted rope was destined to rule all of Asia. The legend is that Alexander the Great came along and sliced right through it. That would not be a valid solution in Knot Theory. There are many other famous knots in history. The endless knot is seen as far back as clay tablets in the Indus Valley and was used in medieval Celtic designs. The Chinese knot is present in Hinduism and Buddhism. Inca tied knots on chords called quipu to track everything from tax to calendars.
Celtic knots are among the most famous examples in Christianity, often found in illuminated manuscripts as intricate, continuous, and interlacing loops. Celtic knots have no discernible beginning or end, symbolizing eternity, particularly God’s eternal love, the soul’s unending nature, and the infinite loop of birth, death, and resurrection. We can see the traces of the cycles of simulated worlds as an endless loop as well. This unbroken cycle has Christ in its center, as the Alpha and Omega, the beginning and the end: “I am the Alpha and the Omega, the First and the Last, the Beginning and the End” (Revelation 22:13). All things begin and end with Christ, although He is beyond time.
Knots tie things together. Therefore, they represent the unity of believers with one another and God. The interconnected loops reflect the theological notion of the Church as the body of Christ, where every member is bound together in spiritual unity and faith: “For just as the body is one and has many members, and all the members of the body, though many, are one body, so it is with Christ” (1 Corinthians 12:12). A knot binds two individuals in the sacrament of marriage. However, the most significant union is depicted in trefoil or triquetra, or the Trinity Knot. This knot features three interlocking arcs symbolizing the Holy Trinity: the Father, the Son, and the Holy Spirit: “Go therefore and make disciples of all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Spirit”(Matthew 28:19). They are three distinct persons yet one Godhead in essence. The continuous design highlights the eternal nature of God, who is without beginning or end. The Holy Trinity is a theological mystery, a complex relationship of unity and diversity. The trefoil is famously difficult to untangle, resembling the nature of God as three in one.
Knots are present everywhere in Christianity. The faith is devotionally shown through the knotted prayer ropes and rosaries. Mary is seen in Catholicism and Orthodoxy as the helper who untangles knots as an act of redemption, mercy and healing: “And I will put enmity between you and the woman, and between your offspring and hers; he will crush your head, and you will strike his heel” (Genesis 3:15). Christian mystics, such as St. John of the Cross or Meister Eckhart, often described the relationship between God and humanity as a kind of mystical knot, focusing on the entangled, complex, and intimate nature of divine-human interaction: “He has sent me to bind up the brokenhearted, to proclaim freedom for the captives and release from darkness for the prisoners” (Isaiah 61:1). The knot signifies the mystery of the divine, unfathomable truths about the whole nature of God we try to discover: “Oh the depths of the riches and wisdom and knowledge of God! How unsearchable are his judgments and how inscrutable his ways” (Romans 11:33).
Unity and Mystery of Knot Theory
The first challenge for mathematicians is to try and figure out how many different types of knots there actually are. One way to classify them is by how many crossings a given knot has. That is, how many times the string passes over or under itself. Some knots look very complicated but can actually be untangled to become much simpler. The key is to find the minimum number of crossings inherent in a given arrangement.
In January 1867, Scottish physicist Peter Guthrie Tait showed off his homemade smoke machine to renowned scientist William Thomson Lord Kelvin. Tait read a paper that claimed that a vortex ring should be eternally stable in an ideal fluid. Intrigued, he set up two wooden boxes containing a kind of toxic mixture of ammonia, sulfuric acid, and salt. When he tapped a towel stretched across the back of each box, the chemical smoke popped out of a circular cutout in perfect rings. Watching, Kelvin was transfixed by the rings. He had been pondering the composition of atoms, a fundamental question of the time, and suddenly saw an answer. He declared that atoms must be made out of vortex rings of ether, an invisible everywhere medium.
Different knots of vortex rings would make different elements. The shape of the Hopf link, the simplest nontrivial link with more than one component, consisting of two circles linked together exactly once and named after Heinz Hopf, explained the double spectral lines of sodium. The simple unknot ring was hydrogen. Tait was skeptical, but as Kelvin’s vortex model of the atom became a leading theory, Tait began investigating knots in earnest and, in his mind, created a periodic table of the elements with every new knot found. The crossing number is a fundamental way to classify knots, representing the minimum number of crossings in the knot’s simplest form, without extra twists or tangles. By hand, Tait systematically discovered knots with increasing crossing numbers: the three-crossing knot (the trefoil), the four-crossing knot (the figure-eight), two distinct five-crossing knots, six six-crossing knots, and seven seven-crossing knots. In knot theory, knots can be combined through operations like the knot sum.
Lord Kelvin’s success had consequences in science. Russian scientist, Mendeleev, published his first periodic table in 1869. The Michelson-Morley experiment sowed seeds of doubt about the ether in 1887. In 1897, JJ Thomson discovered the electron; but Tait was already too deep in Knot Theory. He included his close friend, James Clerk Maxwell, who would become famous for his Maxwell Equation. Maxwell became a knot enthusiast for the rest of his life. With Maxwell’s help, Tait published a list of knots up to seven crossings in 1877. It was the first mathematics paper with the word “knots “in its title.
The considerable labor intensity called for more mathematicians to count knots. One can stick multiple knots together to make a new knot. This is called a composite knot, such as sailing or fishing knots, but some are not decomposable into simpler knots. These are known as prime knots, like the trefoil knot. Since all composite knots are just built from prime knots, people mainly focus on tabulating prime knots. Thomas Kirkman and Charles Little took this task. Together with Tait, they found all 21 eight-crossing knots, 49 nine-crossing knots, and 166 10-crossing knots. This was all done painstakingly by hand.
The structural complexity of knots in three-dimensional space explains how God’s Simulation works through a complex and intelligently designed framework governed by laws or rules that are the fabric of The Simulation. When mathematicians research knots, they find the hidden structure behind seemingly chaotic loops. These findings are relevant for Simulation Creationism because they show reality’s underlying rationality.
Simple loops can form incredibly complex shapes when twisted and entangled. It seems chaotic, but there are precise rules that follow mathematical laws. These rules consider factors like crossings, orientation, and invariants (e.g., the Jones or Alexander polynomials) to determine whether two knots are equivalent or distinct, revealing the underlying structure amidst apparent chaos. The universe is seemingly complex, but it can be downgraded to simple rules that govern simulated environments through mathematical structures. The Bible suggests that God brings order from chaos: “The earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters” (Genesis 1:2). What seems chaotic in this verse was shaped into ordered creation.
Reidemeister Moves
While tabulating knots, Tait saw a central problem in how to tell them apart. The list done by three scientists, Tait, Kirkman, and Little, stood for 75 years without change until a single correction in 1973. In 1927, German mathematician Kurt Reidemeister proved a groundbreaking theorem by showing that any two identical knots can be transformed into each other using only three fundamental operations, now known as Reidemeister Moves. These are the twist, the poke, and the slide, where we move the string from one side of a crossing to the other. The moves are:
- Twist (Type I): adding or removing a loop.
- Poke (Type II): moving one strand across another without breaking the knot.
- Slide (Type III): adjusting a crossing point involving three strands.
These moves proved that some knots are the same if Reidemeister Moves connect them.
Reidemeister moves explain how reality can be transformed or manipulated by God. These specific moves allow for transformations that do not alter the knot’s classification. Thus, we can see this simulated world as part of God’s Simulation, where the Creator manipulates reality through transformations that do not change The Simulation itself. These can be quantum phenomena, cosmic changes, or physical anomalies, often translated as wonders in the mythical sense. It is the idea that God can transform creation without breaking its inherent laws: “The Lord brings death and makes alive; He brings down to the grave and raises up” (1 Samuel 2:6). It shows the transformational power of God, who can use something akin to Reidemeister moves to allow transformations within the boundaries of simulated creation.
Reidemeister moves provide a method by which to determine whether two knots are equivalent by systematically transforming one knot into another. However, identifying the correct sequence of moves can be complex, and failure to find such a sequence does not necessarily mean the knots are different—it may simply mean the right moves have not yet been discovered. But in 1961, mathematician Wolfgang Haken created a computer algorithm that definitely solved the knot equivalence problem for specific cases of distinguishing any knot from the unknot. However, for large or highly complex knots, the algorithm’s runtime would be so extensive that it could exceed the age of the universe, highlighting the computational difficulty of the problem.
In 2001, mathematicians Joel Hass and Jeffrey Lagarias, building on Haken’s work, established an upper bound on the number of Reidemeister moves needed to distinguish a knot from the unknot, providing a systematic method for knot classification. If we check all sequences of Reidemeister moves up to that number, we can prove whether the knot is the unknot. That initial upper bound was 2100,000,000,000n moves, an astronomically large number, far exceeding the number of atoms in the observable universe. Today, the upper bound for Reidemeister moves has been significantly reduced to 236n11, marking a dramatic improvement from earlier estimations. Checking all possible sequences of Reidemeister moves is still unfathomable. For a single crossing knot, this is larger than the number of stars in the observable universe.
In 2011, mathematicians Alexander Coward and Marc Lackenby established an upper bound for the Reidemeister moves required to connect any two knots or links, a major step toward solving the knot equivalence problem. This upper bound involves tetration, a mathematical operation where 2 is iteratively raised to the power of itself multiple times, beginning with 2². The process grows exponentially, ultimately reaching 2 iteratively raised to 21,000,000n, where n represents the crossing number—a measure of the knot’s complexity. The final result is an astronomical value of 101,000,000, ranking among the largest numbers encountered in mathematics.
Just to have a solution is remarkable, given that Turing thought the problem was potentially undecidable only 60 years earlier.
When a knot undergoes a Reidemeister move, its appearance changes, but its identity remains unchanged. This is one of the spectacular features of Knot Theory. We cannot understand knots only by viewing them. They can trick our sight and show themselves differently, although they are not. The knot can look different, although its classification has not changed. It is a challenge of God’s Simulation as well. Similarly, reality may appear different depending on the observer’s perspective, much like how knots can appear transformed without a change in their fundamental properties. A Biblical reflection explains it perfectly: “For now we see only a reflection as in a mirror; then we shall see face to face. Now I know in part; then I shall know fully, even as I am fully known” (1 Corinthians 13:12). Thus, our perception of reality is incomplete or distorted, just as Reidemeister moves can change the appearance of a knot.
The ultimate reality remains unchanged, and we will eventually discover it through God’s grace. It is the process according to which we learn about God’s characteristics and have an opportunity to follow the path of Jesus Christ. Also, the Reidemeister moves are reversible, meaning that a knot can be transformed and then returned to its original form without changing its topological properties. It is the same in The Simulation. A lot of simulated events take place, but the fundamental structure of reality is kept intact: “He has made everything beautiful in its time. He has also set eternity in the human heart, yet no one can fathom what God has done from beginning to end “(Ecclesiastes 3:11). In Ecclesiastes, we are warned that eternal principles are unfathomable, but they preserve the essence of God’s plan.
Knot Invariants: Consistency Amid Complexity
Mathematicians have tabulated 350 million different knots based on their properties, which remain unchanged regardless of how the knot is twisted or tangled. These unchanging properties are called invariants. They will be different for some knots compared to other ones. We can use them as hallmarks of a particular knot. Some knots will share invariants, but if some have different invariants, we know for sure that they are different.
Invariants are fundamental to Knot Theory, as they remain unchanged even when the knot is manipulated. The laws or rules of God’s Simulation are invariants: the laws of physics, the constants of nature, and spiritual truths. These unchanging principles govern the simulated world, even as events unfold in the complex and dynamic way we observe every day. Knot Theory helps us grasp the idea that despite the complexity of the simulated world, it is still governed by unchanging rules: “Jesus Christ is the same yesterday and today and forever” (Hebrews 13:8). This idea of unchanging truth amidst apparent change is echoed in the Bible, where God’s word and laws are described as eternal and unchanging, even though human experience and perception may shift: “Heaven and earth will pass away, but my words will never pass away “(Matthew 24:35). In this verse, Christ reveals that even if the appearance of reality changes, the underlying truth (invariants) remain constant.
The crossing number is an essential invariant of knots. Two knots cannot be identical if they have different crossing numbers. However, the crossing number is surprisingly difficult to calculate. Other variations of the same knot are known as different projections of that knot. Crossing number measures the least number of crossings a knot can have, but it only works for the simplest projection of a knot (reduced form). It is challenging to ensure that a knot is fully reduced. Instead, we can use another invariant that is true right away for all projections of a knot. It will give the same value for both a messy trefoil and a reduced trefoil.
The crossing number can be minimized through simplifications and rearrangements, such as optimization in mathematical systems. Our perceived reality is also optimized. The world we inhabit is a highly optimized system where the crossing number reflecting a compressed form of knot resonates with how a simulated world might be compressed or optimized by the Creator: “The Lord by wisdom founded the earth; by understanding He established the heavens” (Proverbs 3:19). The reality, though complex in appearance, is the product of divine wisdom and efficiency. Using the Knot Theory, we can reduce the apparent complexity and reveal what is happening in our universe.
This means that complexity is a mere illusion. The crossing number is specific to how a knot is projected in 2-dimensional space. By changing the angle or perspective, the number of crossings can appear larger or smaller. Therefore, a seemingly complicated knot can be much simpler when viewed differently. It explains how God positions us as observers. He may hide some things and build on the constraints of human perception. He created us with a limited perspective from which many things in our world seem complex, although, in reality, they are not. We observe things in a restricted way since we need to learn and build our character to be Christ-like. However, there is a vast gap between God’s knowledge and wisdom and the limited perspectives of humans: “For my thoughts are not your thoughts, neither are your ways my ways, declares the Lord. As the heavens are higher than the earth, so are my ways higher than yours and my thoughts than your thoughts” (Isaiah 55:8-9). God’s actions and plans often go beyond what humans can grasp, and what may seem confusing or perplexing to people is part of a greater divine purpose. In Christian theology, there is the concept of mystery, truths beyond human discovery that only God can reveal. Saint Paul considers it to be hidden aspects of God’s plan, hidden for ages but revealed through Jesus Christ: “But we speak the wisdom of God in a mystery, the hidden wisdom which God ordained before the ages for our glory” (1 Corinthians 2:7).
The Simulation may even work exactly as Knot Theory suggests. As the crossing number increases, the knot’s classification becomes more complex, leading to different types of knots with different properties. The crossing number can serve as a threshold that separates simpler from more complex knots. The same logic may be applied to God’s Simulation. God could set specific thresholds of complexity that govern how different simulated worlds or parts of these worlds behave. Certain levels of complexity or interactions might trigger new laws within The Simulation: “He has marked out a boundary on the surface of the waters for a boundary between light and darkness” (Job 26:10).
The first invariant in the Knot Theory is tricolorability, a concept that not only has mathematical significance but also potential implications in broader philosophical and scientific concepts. Tricolorability has two rules. First, we must use at least two colors. Second, at crossings, the three intersecting strands must either be all the same color or all different colors. There are no two-colored crossings and identical knots must match. If one knot is tricolorable and the other one is not, we know they are different knots. Tricolorability is constant across any possible projection of the same knot. It is only needed to prove the knot is not affected by Reidermeister moves.
Not all knots are tricolorable. The ability to assign three colors to a knot within the defined rules depends on the knot’s topology. As such, there are constraints, but also flexibility within those constraints to create a wide range of patterns and structures. We can also look at The Simulation as flexible within the rules and constraints stemming from God’s Singularity. The concept of tricolorability plays a crucial role in understanding the flexible but structured dynamics of The Simulation, where different knots or configurations yield different possibilities, i.e., simulated events and challenges: “In Him we live and move and have our being“ (Acts 17:28).
Tricolorable knots have specific structural properties that are symmetrical, although it is not obvious at first glance. The concept of hidden symmetries help us understand how seemingly complex systems can have underlying simplicity. In the same way, God’s Simulation has hidden symmetries or patterns in The Simulation that we do not immediately perceive: “The secret things belong to the Lord our God, but the things revealed belong to us and to our children forever” (Deuteronomy 29:29). Tricolorability operates on a system of three choices, known as ternary logic. It differs from binary logic in present-day computing, using 0 and 1. Ternary logic represents the more nuanced and flexible system. It gives greater flexibility in designing the rules of the universe. God’s Simulation may well be based on ternary logic, as it can generate a richer diversity of simulated events and challenges. It is a multi-layered logic that could govern the simulated world.
Tricolorability expands into a much more powerful invariant called p-colorability. Here, p can be any prime number besides two. Instead, using colors, mathematicians number each strand with integers between zero and p-1. There are two rules for p-colorability:
- At least two different integers between 0 and p-1 must be used.
- At any crossing, the sum of the integers assigned to the two lower strands, divided by p, must have the same remainder as twice the integer assigned to the top strand, divided by p.
If we test the figure eight-knot for five-colorability, we may assign strand numbers as zero, one, four, and three. At crossings, the numbers must satisfy the modulus condition. For example, at one crossing, (3+4) mod 5 equals 2, proving the knot is five-colorable. This knot is five-colorable and distinct from the unknot. Therefore, p-colorability is a very potent tool. But it still does not cover everythings. Some of the most powerful invariants, the ones that can distinguish between the most unique knots, are polynomials.
The Alexander Polynomial, a significant discovery predating the Reidermeister moves in 1923, is a key element in Knot Theory. Similar to p-colorability, it operates on two fundamental rules. The first rule states that the Alexander polynomial of the unknot is one, while the second allows us to manipulate any single crossing of a knot in three possible positions: forward, backward, and separate. The resulting relationship between the three knots is described by the Alexander polynomial. This polynomial, designed to yield distinct results for as many knots and links as possible, is recursive, allowing for its calculation for increasingly larger knots.
The Alexander polynomial shows how invariant properties may function within The Simulation. Despite the apparent fluidity or changes in how The Simulation is experienced, the core mathematical structure of God’s Simulation remains constant, as dictated by God’s Singularity. The Alexander polynomial may help in revealing consistent structures behind seemingly complex phenomena, in line with the Christian belief that God’s creation is orderly and follows immutable laws: “He has made everything beautiful in its time. He has also set eternity in the human heart; yet no one can fathom what God has done from beginning to end” (Ecclesiastes 3:11). The way the polynomial reduces the complexity of the knot mirrors how God’s Singularity could reduce the apparent complexity of reality into basic rules. In The Simulation, much of the reality experienced by the observers might be superficial, while the underlying structure is hidden within God’s Singularity. The Alexander polynomial may help to understand how certain mathematical tools can uncover the hidden structure of the simulated world, i.e., the divine truths: “It is the glory of God to conceal a matter; to search out a matter is the glory of kings” (Proverbs 25:2).
This polynomial also provides insight into the topological features of knots. It imparts critical information about the structure and connectivity of the knot. It could also be used to show topological and structural properties of God’s Simulation. Namely, The Simulation might have a topological structure, meaning it has networks of events, interactions, space, and time. They can be analyzed using mathematical tools similar to the Alexander polynomial. Such a tool may describe the connections and relationships between different aspects of the simulated reality. This resonates with the belief that God’s creation is not random but follows a divine pattern or structure: “By faith, we understand that the universe was formed at God’s command so that what is seen was not made out of what was visible” (Hebrews 11:3). Simple rules or equations can give rise to emergent properties that are far more complex than the sum of their parts. The idea of emergence is foundational for Simulation Creationism. Simple rules stemming from God’s Singularity might give rise to complex phenomena like consciousness, natural laws, and religious structures. God’s creation, built on divine principles, allows life, order, and complexity to emerge.
The Alexander polynomial stood unchanged for over sixty years as the knot invariant of choice. However, in 1984, mathematician Vaughan Jones had been working on a type of algebra for statistical mechanics when he realized his work resembled a series of equations in Knot Theory. The Jones polynomial is like Alexander but has a more specific equation for the second rule that distinguishes many more knots. The first new polynomial invariant kicked up a fervor in Knot Theory. In a matter of months, six mathematicians, each independently, found an improved version of his polynomial with two variables instead of one. Their combined findings are known as Homfly Polynomial. None of these invariants works alone. Knots have dozens of invariants, which, when taken together, uniquely identify them.
The Jones polynomial is a more refined tool than the Alexander polynomial since it captures more subtle features of a knot’s structure. A similar mathematical tool is required to analyze and decode the hidden structure of the simulated world. The mathematical foundations of The Simulation may be far more intricate than they appear. A tool similar to the Jones polynomial could help uncover the hidden architecture. Consequently, we may reach the fundamental principles that are consistent and unchanged, as an invariant should be: “He established the earth upon its foundations so that it will not totter forever and ever” (Psalm 104:5).
It had taken decades of work to tabulate all 249 prime knots up to ten crossings. The task of identifying knots with 11 crossings remained unattempted until John Conway successfully cataloged all 552 prime knots at this level, completing the tabulation by hand. In 1983, Dowker and Thistlewaite developed a computer algorithm to enumerate all knots with 12 and 13 crossings. Later, in collaboration with Hoste and Weeks, they extended this work to include knots with 14, 15, and 16 crossings. This method, still in use today, employs computer algorithms to generate all possible knots and uses invariants to efficiently eliminate duplicates.
In 2020, mathematician Ben Burton single-handedly tabulated all 17, 18, and 19 crossing knots, bringing the total number of known prime knots to 352,152,252. His project was so computationally intensive that several hundred computers had to run for months before obtaining the final number. The hardest part of knot tabulation is counting up every knot and then carefully eliminating duplicates. If we want to generate them, we can make alternating knots, i.e., knots with crossings that alternate over and under. This computation is much easier, although it leaves out most knots. In 2007, researchers used this method to identify alternating knots with up to 24 crossings. Overall, the total number of knots, including all types, is estimated at 159,965,097,353.
Knot theorists have developed many other ways to classify knots with algorithms, equations, or geometric descriptions. One way is to flatten out a knot and imagine it as a complicated roadway with overpasses and underpasses. They describe the route numerically, writing which underpass matches each overpass. None of these approaches is perfect. Sometimes, one mathematical solution can describe several different knots, making that particular method not enough to distinguish them.
Knot Theory, Higher Dimensions, and God’s Simulation
In their unique approach, mathematicians extend the concept of knots into higher dimensions. In classical Knot Theory, a knot is a closed loop embedded in three-dimensional space that cannot be untangled without cutting it. This unique extension into higher dimensions opens up new possibilities and challenges in the study of knots. In 4D (three-dimensional space plus time), mathematicians consider hollow knotted spheres instead of knotting loops of 2-dimensional string. These spheres are analyzed by examining the 3D cross-sections produced when slicing them at various points. A slice through an unknotted sphere would be the trivial knot, i.e., a simple loop. However, slicing a knotted sphere can result in one of the known prime knots, depending on the slice’s position. Thus, 4D knots provide greater flexibility, offering new insights into their interactions and properties within higher-dimensional spaces. In mathematical terms, a knot in 3D can be seen as a 2D surface in 4D. These are called Seifert surfaces, and they can be studied through knot invariants and other tools to understand how higher-dimensional spaces influence the structure of 3D knots. It suggests that complex phenomena in 3D (such as natural laws) might actually be simpler in the higher-dimensional source of God’s Simulation.
The flexibility of untangling knots in 4D is how the Creator has access to dimensions and processes beyond our comprehension and how God’s Singularity easily manipulates The Simulation. A higher-dimensional knot can be projected into lower dimensions, such as 3D or 2D space. It simplifies some properties of the knot while also allowing insights about the higher-dimensional structure from its lower-dimensional projection. Such an understanding of knots is very helpful in comprehending God’s Simulation. Our world might be a lower-dimensional projection of a higher-dimensional reality. Just as a 4D knot projected into 3D space reveals only part of its structure, the reality we perceive is a partial and simplified version of a more complex divine system. We are given bits and pieces of higher dimensions in a way we can fathom, but it also shows us there are higher truths out there.
In higher dimensions, the concept of knot complements, spaces that remain after the knot is removed from the 3D space in which it is embedded, plays a crucial role. By studying these spaces, mathematicians gain a deeper understanding of the topology of both the knot and the surrounding space. This understanding extends to manifolds of higher dimensions, providing insights into the structure of 4-manifolds, which are important objects in higher-dimensional topology and string theory.
Knot Theory has found its way into the quantum world, too. Quantum particles do not typically follow the shortest path from A to B. Instead, they can take any number of strange loops and zigzags. To try and predict the behavior of things on the quantum scale, physicists need to consider all the possible routes these particles could take. Knot theory, which aims to reduce complex entanglements into simpler and more fundamental forms, helps solve that problem. Although Knot Theory does not fully encapsulate all aspects of quantum phenomena, its numerical approaches to classifying knots provide valuable approximations for predicting the most probable quantum paths. This information helps physicists understand the quantum world and how to harness it to build things like quantum computers.
Higher-dimensional Knot Theory is linked to quantum mechanics, particularly through quantum field theories and quantum invariants derived from knots. These quantum features of knots provide insights into the underlying mathematical structures that may suggest a discrete nature of space-time in higher dimensions, although they do not directly establish such discreteness. Further elaboration and evidence are needed to connect knot theory with the fundamental properties of space-time. Thus, the quantum nature of reality in our world exists as a working mode of God’s Singularity. Quantum theory allows for phenomena like superposition and entanglement. Still, the findings from higher-dimensional knots suggests that the structure of reality is multifaceted and flexible at the quantum level, meaning that God may use quantum mechanics as a control of The Simulation: “Great is our Lord and mighty in power; His understanding has no limit” (Psalm 147:5).
Knots in DNA
Knot theory is also critical to biological processes that saved millions of lives. Bacterial DNA consists of a single loop of the double helix molecule. This circular shape often results in knotted or entangled configurations during replication, preventing the bacteria from properly segregating their DNA into two cells. Thus, they have type II topoisomerase enzyme, which snips and reconnects the DNA. This turns their linked DNA back into an unlink so they can replicate cleanly. This is the way some of the most common antibiotics in the world, called quinolones, operate. Knot theory helps model biological phenomena, as DNA strands often behave like mathematical knots. In Simulation Creationism, DNA can be seen as an algorithm that governs the emergence and development of life. The fact that DNA behaves like a knot, with highly regulated untangling processes, suggests a level of order stemming from God’s Singularity: “For you created my inmost being; you knit me together in my mother’s womb “(Psalm 139:13). In fact, DNA, with its intricate structure and role for life, is the most direct evidence of a simulated design. God’s Singularity optimizes DNA to create a coherent and self-sustaining reality, carefully crafted with wisdom and intentionality.
Knot theory may even include quantum mechanics in the behavior of DNA, particularly how protons tunnel between different positions in the molecule, potentially influencing mutation and replication processes. DNA’s quantum behavior might reflect the discrete nature of God’s Singularity, where quantum rules are a byproduct of the planned structure of the simulated world: “The wind blows wherever it pleases. You hear its sound, but you cannot tell where it comes from or where it is going. So it is with everyone born of the Spirit”(John 3:8).
Human DNA, while not circular, is long enough to get into tangles. Each cell in our bodies contains two meters of DNA. When it inevitably tangles, human Type II topoisomerase comes to make crossing changes. The human version of the enzyme is different enough from the bacterial version, unaffected by antibiotics. But human topoisomerase is sometimes intentionally inhibited. This stops replication and kills cells, predominantly the rapidly dividing cancer cells. It is one of the most common forms of chemotherapy. Biologists needed Knot Theory to first understand the mechanism of Type II topoisomerase. They decreased the crossing number of knots in DNA two at a time and realized it had to be cutting and rejoining entire double strands of DNA. There are many other obscure topoisomerases that act on DNA. Knot theory is used to analyze the knots they tie or untie and how they operate as a result. It is not just DNA since 1% of all proteins have various knots in their fundamental structure. If they get misknotted, they malfunction. However, the potential of knot theory in accurately identifying and understanding these knots gives us hope. It could lead to a better understanding of these proteins’ mechanisms, as well as how to potentially repair or utilize them, offering a ray of hope in the fight against diseases.
When DNA becomes knotted during cellular processes, topoisomerase intervenes to cut and untangle the DNA, ensuring that it will continue to function, similar to Reidemeister Moves. It is also God’s intervention in resolving complex issues within The Simulation. The knotting and untying of DNA is how the Lord adjusts the internal structure to maintain its function: “For I know the plans I have for you, declares the Lord, plans to prosper you and not to harm you, plans to give you hope and a future “(Jeremiah 29:11).
