🪢Knot Theory Unit 1 – Knot Theory Basics: Intro & Definitions

Key Concepts and Definitions

• Knot theory studies mathematical knots, closed loops in 3-dimensional space that do not intersect themselves
• A knot is a single closed curve that does not intersect itself and cannot be untangled to form a simple loop without cutting
• Links are collections of knots that may be interlocked or linked together but do not intersect each other
• Knot equivalence states that two knots are equivalent if one can be continuously deformed into the other without passing through itself (ambient isotopy)
  • Equivalent knots are considered the same knot type and share invariant properties
• Knot invariants are properties of knots that remain unchanged under continuous deformations (crossing number, tricolorability)
• The unknot, also known as the trivial knot, is the simplest knot that forms a simple closed loop
• Knot diagrams are 2-dimensional representations of knots using a projection onto a plane with over/under crossing information at each intersection
• Reidemeister moves are local changes to a knot diagram that do not change the underlying knot type (three types: twist, poke, slide)

Historical Background

• Knot theory has roots in the 19th century work of physicists and mathematicians studying the behavior of strings and ropes
• Early investigations focused on classifying and tabulating knots, such as the work of Peter Guthrie Tait in creating the first knot tables
• In the 1920s, Kurt Reidemeister introduced the Reidemeister moves, establishing a foundation for the modern study of knot equivalence
• The development of knot polynomials in the 1980s, such as the Jones polynomial and HOMFLY-PT polynomial, provided powerful new tools for distinguishing knots
• Recent decades have seen the application of knot theory to diverse fields, including physics, chemistry, and biology
• The study of knots has led to the development of related areas, such as braid theory and the theory of 3-manifolds
• Knot theory has benefited from the use of computational methods and the development of knot databases and software tools

Types of Knots

• Torus knots are knots that can be drawn on the surface of a torus without self-intersection (trefoil knot, cinquefoil knot)
• Satellite knots are constructed by taking a knot (the companion) and replacing a neighborhood of each point with another knot (the pattern)
• Hyperbolic knots are knots whose complement (the space obtained by removing the knot from the 3-sphere) admits a hyperbolic geometry
• Alternating knots have a diagram where the crossings alternate between over and under as one travels along the knot
• Non-alternating knots do not admit an alternating diagram and often have more complex properties and invariants
• Prime knots cannot be decomposed as the connected sum of two non-trivial knots
  • Composite knots are formed by taking the connected sum of two or more prime knots
• Amphichiral knots are equivalent to their mirror image, while chiral knots are not

Basic Knot Operations

• Connected sum takes two knots and joins them together by removing a small arc from each knot and connecting the endpoints
  • The connected sum is denoted by $K_1 \# K_2$ and is commutative and associative
• Knot multiplication involves placing one knot inside a regular neighborhood of another knot, creating a satellite knot
• Cabling is a specific type of satellite construction where the pattern knot is a torus knot wrapping around the companion knot
• Mutation alters a knot by cutting out a tangle (a region of the knot diagram) and regluing it with a rotation or reflection
  • Mutant knots share many invariants but can sometimes be distinguished using more sophisticated techniques
• Unknotting operations, such as a crossing change or a $\Delta$-move, can be used to simplify a knot diagram or determine the unknotting number
• Band surgery involves cutting the knot along a band and regluing the ends with a half-twist, potentially changing the knot type
• Taking the branched double cover of a knot complement produces a 3-manifold, linking knot theory to the study of 3-dimensional spaces

Mathematical Representations

• Knot diagrams are 2-dimensional projections of knots onto a plane, with over/under crossing information at each intersection
• Reidemeister moves are local diagram changes that do not alter the underlying knot type (twist, poke, slide)
• Knot polynomials are algebraic invariants that assign a polynomial to each knot (Jones polynomial, HOMFLY-PT polynomial, Alexander polynomial)
  • These polynomials can distinguish between different knot types and provide information about knot properties
• The knot group is the fundamental group of the knot complement and encodes information about the knot's topology
  • Knot groups can be presented using generators and relations derived from the knot diagram (Wirtinger presentation)
• The Alexander matrix is a matrix of polynomials derived from the knot group and gives rise to the Alexander polynomial
• Seifert surfaces are oriented surfaces bounded by a knot, used to compute the knot genus and define the Seifert matrix
• The Seifert matrix encodes linking information between curves on the Seifert surface and is used to define the Alexander polynomial and signature invariants

Applications in Other Fields

• In physics, knots appear as stable configurations in classical field theories and as topological defects in condensed matter systems
  • Knotted vortices in fluids and knotted dislocations in crystals are examples of physical knots
• Knot theory is used in the study of DNA and protein structure, where molecules can form knotted or linked configurations
  • Knot invariants are employed to classify and understand the properties of these biological knots
• In chemistry, molecular knots are synthesized and studied for their unique properties and potential applications (molecular machines, catalysts)
• Knots and links are used in the design and analysis of quantum algorithms and quantum error-correcting codes
• Knotted graphs and networks appear in the study of complex systems, such as social networks and neural networks
• Knot theory has applications in computer graphics and animation, where knots are used to model and simulate realistic rope and cloth behavior
• In mathematics, knot theory has connections to various areas, including algebraic topology, geometric topology, and representation theory

Common Misconceptions

• Not all closed loops in 3D space are considered knots in the mathematical sense, as knots must not intersect themselves
• Knot equivalence is not the same as visual similarity, as equivalent knots may have very different diagrams
  • Determining knot equivalence can be a challenging problem, and there is no general algorithm for deciding if two knots are equivalent
• The unknot is a true knot in knot theory, even though it may seem "unknotted" in the colloquial sense
• Knot diagrams are not unique, as different diagrams may represent the same knot type
  • Reidemeister moves can be used to relate different diagrams of the same knot
• Knot polynomials are not complete invariants, meaning that non-equivalent knots may have the same polynomial values
  • However, knot polynomials are still powerful tools for distinguishing between many knots
• The crossing number of a knot is not always equal to the minimum number of crossings in a diagram, as some knots admit diagrams with fewer crossings than their true crossing number
• Not all properties of knots are invariant under continuous deformations (color, texture, thickness), and it is essential to distinguish between invariant and non-invariant properties

Further Exploration

• Higher-dimensional knot theory studies knotted spheres in 4-dimensional space and higher dimensions
  • 2-knots are knotted 2-spheres in 4-space and have unique properties compared to classical 1-knots
• Virtual knot theory extends the concept of knots to allow for virtual crossings, which do not have a fixed over/under relationship
  • Virtual knots can be used to model knots in thickened surfaces and have applications in quantum topology
• Legendrian and transverse knot theory study knots in contact 3-manifolds, where the knots are constrained to be tangent or transverse to a contact structure
• The study of finite-type invariants, also known as Vassiliev invariants, provides a unified framework for understanding and classifying knot invariants
• Khovanov homology is a powerful knot invariant that categorifies the Jones polynomial and has deep connections to representation theory and mathematical physics
• The study of knots in 3-manifolds other than the 3-sphere, such as lens spaces or hyperbolic 3-manifolds, reveals new phenomena and connections to geometric topology
• The relationship between knots and 3-manifolds is a rich area of study, with knots playing a key role in the classification and understanding of 3-dimensional spaces
• Knot theory has potential applications in the study of topological quantum computation, where knots and braids are used to model and manipulate quantum states