Knot theory

Knot theory is a branch of mathematics that studies mathematical knots, which are loops in three-dimensional space that do not intersect themselves. It is a part of the field of topology, specifically dealing with the properties of these loops that remain unchanged through continuous deformations, such as stretching, twisting, and bending, but not cutting or gluing. In knot theory, a "knot" is defined as an embedded circle in three-dimensional Euclidean space \( \mathbb{R}^3 \).

Knot invariants are properties or quantities associated with a knot that remain unchanged under certain transformations, such as knot deformation (rearranging the knot without cutting it). Knot invariants are essential in the study of knot theory, a branch of topology that explores the mathematical properties of knots and their classifications. There are several types of knot invariants, each providing different insights into the structure and characteristics of knots.

Knot operations refer to methods used in the field of knot theory, a branch of topology in mathematics that studies the properties and classifications of knots. A knot is defined as a closed loop in three-dimensional space that does not intersect itself, akin to a tangled piece of string. Knot operations are techniques that allow mathematicians to manipulate these knots to study their properties, relationships, and classifications.

Knot theory is a branch of mathematics that studies mathematical knots, which are defined as embeddings of a circle in three-dimensional space. Knot theory investigates properties of these knots, such as their classification, properties, and invariants. A "stub" in this context typically refers to a brief or incomplete entry or overview, often found in wikis or databases, that provides only basic information on a topic.

Knots and links are concepts from the field of topology, a branch of mathematics that studies the properties of space that are preserved under continuous transformations. ### Knots: A **knot** is essentially a closed loop in three-dimensional space that does not intersect itself. To understand knots, imagine taking a piece of string, tying it into a loop, and then slicing through space without cutting the string apart.

A 2-bridge knot is a specific type of knot in the field of topology, particularly in the study of knot theory. It is characterized by having a diagram that can be represented with only two bridges, or arcs, connecting the points where the knot crosses itself.

Alexander's theorem, often associated with the mathematician James Waddell Alexander II, refers to several concepts in mathematics, depending on the context. Here are a couple of notable ones: 1. **Alexander's Theorem in Topology**: This theorem relates to the concept of homeomorphisms of topological spaces. It states that every simple closed curve in the plane divides the plane into an "inside" and an "outside," forming distinct regions.

The Alexander matrix, often used in the study of knot theory, is a specific type of matrix associated with a knot or link. It plays a crucial role in analyzing the topology of knots and can be used to derive the Alexander polynomial, an important invariant of knots. The Alexander matrix is constructed from the following steps: 1. **Representation**: Start with a knot or link diagram. From this diagram, choose a triangular decomposition of the knot/link complement.

An alternating planar algebra is a mathematical structure that arises in the study of planar algebras, a concept introduced by Vaughan Jones in the context of knot theory and operator algebras. Planar algebras are a combinatorial framework that allows for the abstract representation of algebraic structures using diagrams drawn on the plane. They generalize the notion of tensor products and can describe a variety of algebraic objects, including link invariants, quantum groups, and more.

Arithmetic topology is an emerging field at the intersection of arithmetic geometry and topology. It brings together concepts from both disciplines to study the topological properties of spaces that arise in number theory and algebraic geometry, particularly focusing on the properties of various kinds of schemes and their associated topological spaces. A prominent theme in arithmetic topology is the study of the relationships between algebraic objects (like varieties) and their topological counterparts.

The average crossing number of a graph is a concept from graph theory that relates to the arrangement of edges in a graph when drawn in the plane. Specifically, it quantifies the average number of crossings that occur when edges are drawn between vertices. ### Key Points: 1. **Graph Drawing**: When a graph is drawn on a plane, edges might cross each other. A crossing occurs whenever two edges intersect at a point that is not a vertex.

The Birman–Wenzl algebra, often denoted as \( BW_n \), is an algebraic structure that arises in the study of knot theory, representation theory, and those interactions with combinatorics. It is named after Joan Birman and Hans Wenzl, who introduced it in the context of their work on braids and coloring of knots.

The braid group is a mathematical structure that arises in the study of braids, which can be visualized as strands intertwined in a particular way. It is a fundamental concept in the fields of topology, algebra, and mathematical physics.

Chirality in mathematics refers to a property of a geometric object that is not superimposable on its mirror image. This concept is derived from the field of topology, which studies properties of space that are preserved under continuous transformations. In a more formal mathematical sense, chirality can be defined in relation to certain structures or shapes, particularly in: 1. **Geometric Objects**: For example, the left and right hands are classic examples of chiral objects.

In mathematics, "Clasper" refers to a tool or device used in the study of knot theory and low-dimensional topology. Specifically, a clasper is a specific type of graph-like structure that can be used to manipulate knots and links. Claspers can be thought of as a generalization of the notion of a "knot insertion" and are used to define operations that can alter the topology of a knot or link.

The Cyclic Surgery Theorem is a result in the field of topology, particularly in the study of 3-manifolds. It relates to the behavior of 3-manifolds under certain types of surgeries, which are operations that alter the topology of a manifold. More specifically, this theorem is often discussed in the context of hyperbolic 3-manifolds.

Dehornoy order is a specific ordering on the set of braids, which is primarily used in the study of braids and their algebraic properties. Named after the mathematician Patrick Dehornoy, the Dehornoy order provides a way to compare braids based on their geometric and combinatorial structure. In the context of braids, the Dehornoy order can be defined with the help of certain moves and words that represent braids.

Dowker–Thistlethwaite notation is a method used in knot theory to represent knots and links in a compact form. This notation encodes information about a knot's crossings and their order, facilitating the study of knot properties and transformations. In Dowker–Thistlethwaite notation, a knot is represented by a sequence of integers, which are derived from a specific way of traversing the knot diagram.

Fox n-coloring is a mathematical concept related to graph theory, specifically focusing on the study of graphs and their colorings. It is named after mathematician Jonathan Fox. In general, the Fox n-coloring of a graph assigns colors to the vertices of the graph such that certain conditions are met, allowing for the examination of various properties and structures within the graph.

The term "Free Loop" can refer to different concepts depending on the context. Here are a few interpretations: 1. **Software Development**: In programming or software design, a "free loop" might refer to a loop that does not have predefined limits, allowing for iteration based on dynamic conditions rather than fixed iterations.

The Fáry–Milnor theorem is a result in the field of geometric topology, specifically concerning the properties of simple closed curves in three-dimensional Euclidean space. The theorem states that every simple closed curve in \(\mathbb{R}^3\) can be represented as a polygonal curve (a finite concatenation of straight line segments) with a finite number of vertices.

Gauss notation, often referred to as "big O" notation, is a mathematical notation used to describe the asymptotic behavior of functions. It provides a way to express how the output of a function grows relative to its input as the input approaches a particular value, commonly infinity. The term "Gauss notation" is not widely used; it is more commonly known as "asymptotic notation" or "Big O notation.

The Gordon–Luecke theorem is a result in the field of geometry and topology, specifically in the area concerning the classification of certain knots in three-dimensional space. The theorem establishes a criterion for determining when two nontrivial knots in \( S^3 \) (the three-dimensional sphere) are equivalent or can be transformed into one another through a process known as knot concordance.

Knot theory is a branch of topology that studies mathematical knots, which are defined as closed, non-intersecting loops in three-dimensional space. The history of knot theory can be traced through several key developments and figures: ### Early Developments - **Ancient Civilization:** The earliest practical understanding of knots is found in various cultures, where knots played a significant role in fishing, navigation, and clothing.

Hyperbolic volume typically refers to the volume of a three-dimensional hyperbolic manifold, which is a type of manifold that exhibits hyperbolic geometry. In hyperbolic geometry, the space is negatively curved, in contrast to Euclidean geometry, which is flat, and spherical geometry, which is positively curved. The concept of hyperbolic volume is most often studied in the context of three-dimensional hyperbolic manifolds.

In topology, the complement of a knot refers to the space that remains when the knot is removed from the three-dimensional space.

Knot energy refers to the energy associated with the configuration or shape of a knot in a physical system, usually relating to the fields of physics, mathematics, or materials science. It primarily considers the work done against forces (such as tension) to create or maintain a knot. In a more specific context, knot energy can be applied to biological systems, like DNA, where the energy configuration of a knotted or constrained DNA molecule can affect its biological functions and stability.

Knot tabulation is a method used in knot theory, a branch of topology that studies mathematical knots. This technique involves creating a systematic list (or table) of knots and links based on specific characteristics such as their knot type, crossing number, and other invariants. The purpose of knot tabulation is to organize and classify knots for easy reference, comparison, and study.

Knot thickness typically refers to the measurement of the thickness of a knot in a material, such as rope, cord, or string. In the context of textiles, knots can affect the overall thickness of the material, which can influence its flexibility, strength, and appearance. For example, in fishing lines, the thickness of a knot can impact how it moves through water and its chances of snagging.

"Knots Unravelled" appears to refer to a book and educational resource by Dr. Vanessa M. H. Lin, which explores the intricate and fascinating world of knots from both a mathematical and a practical perspective. The book delves into the history, theory, and applications of knots in various fields, such as science, engineering, and everyday life. Additionally, it includes discussions on knot theory, which is a branch of topology that studies mathematical knots.

The "lamp cord trick" typically refers to a method often used in the context of magic or illusions. In this trick, a piece of electrical cord (like a lamp cord) is manipulated in such a way that it appears to do something magical or impossible—such as moving on its own or being tied and untied without apparent effort.

Linkless embedding is a concept in the field of data representation and machine learning, particularly relevant in the context of graph-based models and natural language processing. The term often relates to the way certain types of information can be represented without relying on explicit connections or links between data points, such as in traditional graph structures. In the context of machine learning: 1. **Graph Representation**: Many machine learning tasks depend on nodes (data points) and edges (connections between nodes).

Knot theory is a branch of mathematics that studies knots, their properties, and the various ways they can be manipulated and classified. Here is a list of topics within knot theory: 1. **Basic Concepts** - Knots and links: Definitions and examples - Open and closed knots - Tangles - Reidemeister moves - Knot diagrams 2. **Knot Invariants** - Fundamental group - Knot polynomials (e.g.

The list of mathematical knots and links refers to the classification and naming of different types of knots and links studied in the field of topology, particularly in knot theory. Knots are closed curves in three-dimensional space that do not intersect themselves, and links are collections of two or more knots that may or may not be interlinked. Here are some commonly recognized knots and links: ### Knots 1. **Unknot**: The simplest knot, which is equivalent to a simple loop.

A list of prime knots refers to a classification of knots in the field of topology, specifically knot theory. In knot theory, a knot is typically defined as a loop in three-dimensional space that does not intersect itself. Knots can be composed in various ways, and when a knot cannot be decomposed into simpler knots (i.e., cannot be divided into two non-trivial knots that are linked together), it is referred to as a "prime knot.

Loop representation is a conceptual and mathematical framework used primarily in the context of quantum gauge theories and quantum gravity. It emerges from attempts to quantize these theories, especially when dealing with the complexities arising from gauge invariance and non-abelian gauge groups. Here’s an overview of its significance and structure: ### Overview of Loop Representation 1. **Gauge Theories**: Infield theories, gauge symmetries, and associated gauge groups play a vital role.

The Milnor conjecture, proposed by John Milnor in the 1980s, is related to the topology of smooth manifolds and stems from the study of smooth structures on high-dimensional spheres. Specifically, it concerns the relationship between the topology of some manifolds and certain algebraic invariants derived from their smooth structures.

The Milnor map arises in the study of the topology of manifolds, particularly in the context of smooth invariants and characteristic classes. Named after John Milnor, it provides a way to analyze the relationships between different types of differentiable structures on manifolds.

Möbius energy is a concept from theoretical physics that describes a type of conserved energy associated with certain symmetries in systems, particularly in quantum mechanics and field theory. The term "Möbius" often refers to structures or phenomena related to the Möbius strip, a non-orientable surface with interesting topological properties.

Petal projection is a type of map projection used to visualize geographical data in a way that emphasizes certain features or regions, often in thematic mapping contexts. It derives its name from its visual resemblance to petals of a flower, as the projection often extends outward in a radial fashion, resembling petals surrounding a central point.

Physical knot theory is an interdisciplinary field that combines concepts from mathematics, physics, and biology to study the properties and behaviors of knots and links in various physical contexts. This area of research looks at how knots form, evolve, and interact in different physical systems, using tools from topology and applying them to real-world phenomena.

Planar algebra is a mathematical structure that arises in the study of operator algebras and three-dimensional topology. It was introduced by Vaughan Jones in the context of his work on knot theory and nontrivial solutions to the Jones polynomial. Planar algebras provide a framework for understanding the relationship between combinatorial structures, algebraic objects, and topological phenomena. In essence, a planar algebra consists of a collection of vector spaces parameterized by non-negative integers, typically with a specified multiplication operation.

Quadrisecant is a term that typically refers to a numerical method used for finding roots of equations. It is a specific type of secant method that operates using a modified approach to accommodate the scenarios where more than two points are available or necessary. In the context of numerical methods, the secant method itself approximates the roots of a function by using two initial guesses and forming a secant line.

In the context of quantum mechanics and quantum field theory, the term "quantum invariant" generally refers to a property or quantity that remains unchanged under certain transformations or changes in the system. Here are some key points regarding quantum invariants: 1. **Symmetry and Invariance**: Quantum invariants often relate to symmetries in physical systems.

Racks and quandles are concepts from the field of algebra, particularly in the study of knot theory and algebraic structures.

Regular isotopy is a concept from the field of topology, particularly in the study of knots and links. It refers to a continuous transformation of a knot or link in three-dimensional space that can be performed without cutting the string, self-intersecting, or passing through itself.

A Seifert surface is a surface used in the field of topology, particularly in the study of knots and links in three-dimensional space. Named after Herbert Seifert, these surfaces are oriented surfaces that are bounded by a given link in the three-dimensional sphere \( S^3 \). The key properties and characteristics of Seifert surfaces include: 1. **Boundary**: The boundary of a Seifert surface is a link in \( S^3 \).

"Skein relation" refers to a concept in the study of knots and links within the field of topology, specifically in knot theory. Skein relations are equations that express the relationships between different knots or links under certain conditions. These relations are often used in the computation of polynomial invariants of knots and links, such as the Jones polynomial or the HOMFLY-PT polynomial. The basic idea behind skein relations is to define a knot or link in terms of simpler components.

The slice genus is a concept from the field of topology, specifically in the study of 4-manifolds and knot theory. It is defined as follows: 1. **Knot Theory Context**: In knot theory, the slice genus of a knot in 3-dimensional space is a measure of how "simple" the knot is in terms of being able to be represented as the boundary of a smooth, oriented surface in a 4-dimensional space.

The Tait conjectures, proposed by the Scottish mathematician Peter Guthrie Tait in the late 19th century, relate to the field of knot theory, a branch of topology. Tait conjectured that there are specific relationships between the number of crossings in a knot diagram and its properties, particularly concerning its link or knot type.

The Knot Atlas is a wedding planning tool and resource offered by The Knot, a popular wedding planning website. The Atlas provides couples with personalized wedding ideas and inspiration by showcasing various venues, vendors, and wedding styles based on location. It helps users explore options tailored to their preferences, including different themes, budgets, and settings, making the wedding planning process more organized and efficient.

The Unknotting Problem is a well-known problem in the field of topology, particularly in knot theory, which is a branch of mathematics that studies the properties and classifications of knots. The problem can be stated as follows: **Problem Statement**: Given a knot (a closed loop in three-dimensional space that does not intersect itself), determine whether the knot is equivalent to an "unknotted" loop (a simple, non-intersecting circle).

The Volume Conjecture is a mathematical hypothesis related to the field of knot theory and hyperbolic geometry. It proposes a deep connection between the volumes of hyperbolic 3-manifolds and quantum invariants of knots, specifically those derived from a quantum invariant known as the Kauffman polynomial or the colored Jones polynomial.

The vortex theory of the atom, often associated with the work of 19th-century physicist William Thomson (also known as Lord Kelvin), proposes that atoms are not solid, indivisible particles but rather are composed of swirling vortices in the aether. According to this theory, these vortices would represent the fundamental particles of matter, with their motions and interactions giving rise to the properties of atoms and molecules.

Willerton's fish (scientific name: *Sicyopterus williardsoni*) is a species of freshwater fish belonging to the family Gobiidae. It is notable for its unique characteristics, including its small size and adaptations to a specific habitat. Willerton's fish is primarily found in the streams and rivers of tropical regions, often inhabiting areas with rocky substrates and fast-flowing waters.

The Wirtinger presentation is a method used in algebraic topology, particularly in the study of the fundamental group of a braid or a link in three-dimensional space. It was introduced by Wilhelm Wirtinger in the early 20th century. The Wirtinger presentation gives a way to describe the fundamental group of a knot or link by associating it with a specific set of generators and a set of relations derived from a diagram of the knot or link.

"Writhe" can refer to several different concepts depending on the context: 1. **Biological Context**: In biology, "writhe" often describes the movement of animals, particularly when they are twisting or contorting their bodies in reaction to pain or distress. For example, snakes or worms might writhe on the ground.