🪢Knot Theory Unit 3 – Knot Equivalence and Knot Invariants

Key Concepts and Definitions

• Knot theory studies mathematical knots, which are closed curves in three-dimensional space that do not intersect themselves
• A knot is a single closed loop, while a link consists of multiple interlocking closed loops
• Knot diagrams are two-dimensional representations of knots using a projection onto a plane with over/under crossing information
• Knot equivalence determines whether two knots can be continuously deformed into each other without cutting or passing through itself
• Knot invariants are properties or values assigned to knots that remain unchanged under knot equivalence, helping distinguish between different knots
  • Examples of knot invariants include tricolorability, knot polynomials (Alexander, Jones, HOMFLY-PT), and crossing number
• Ambient isotopy is a continuous deformation of the surrounding space that carries one knot into another, preserving knot type
• The unknot, also known as the trivial knot, is the simplest knot with no crossings

Types of Knots and Their Properties

• 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 its tubular neighborhood with another knot (the pattern)
• Hyperbolic knots are knots whose complement has a hyperbolic geometry structure (figure-eight knot)
• Alternating knots have diagrams where the crossings alternate between over and under as one travels along the knot
• Non-alternating knots have diagrams where the crossings do not alternate consistently (Perko pair)
• Prime knots cannot be decomposed into simpler knots by cutting along a sphere (trefoil knot, figure-eight knot)
  • Composite knots are formed by connecting two or more prime knots (granny knot, square knot)
• Amphichiral knots are equivalent to their mirror images, while chiral knots are not

Knot Equivalence: Reidemeister Moves

• Reidemeister moves are local diagram moves that do not change the knot type, allowing for knot equivalence
• Reidemeister move I (twist) adds or removes a twist in the diagram
• Reidemeister move II (poke) adds or removes two crossings in a strand
• Reidemeister move III (slide) slides a strand over or under a crossing
• Any two diagrams of equivalent knots can be related by a finite sequence of Reidemeister moves
• Reidemeister theorem states that two knots are equivalent if and only if their diagrams are related by a finite sequence of Reidemeister moves
• Tricolorability is preserved under Reidemeister moves, making it a knot invariant
• Applying Reidemeister moves to simplify a knot diagram can help determine its knot type and calculate invariants

Introduction to Knot Invariants

• Knot invariants are properties or values assigned to knots that remain unchanged under knot equivalence
• Invariants help distinguish between different knots and determine if two knots are equivalent
• Some invariants are complete, meaning they can distinguish all non-equivalent knots, while others are incomplete and may assign the same value to different knots
• Tricolorability is a binary invariant that checks if a knot diagram can be colored with three colors while satisfying certain rules at crossings
• Knot polynomials are invariants that assign a polynomial to each knot (Alexander polynomial, Jones polynomial, HOMFLY-PT polynomial)
  • These polynomials encode information about the knot and can distinguish many non-equivalent knots
• Crossing number is the minimum number of crossings in any diagram of a knot, serving as a simple invariant
• Unknotting number is the minimum number of crossings that need to be changed to turn a knot into the unknot

Common Knot Invariants and Their Calculations

• Tricolorability: Assign colors to arcs in a knot diagram following rules at crossings; knot is tricolorable if coloring succeeds
• Alexander polynomial: Computed using the Alexander matrix derived from a knot diagram; encodes knot symmetries and genus
  • Calculated by labeling arcs, setting up relations at crossings, and computing determinant of the resulting matrix
• Jones polynomial: Computed using the Kauffman bracket, a state sum that satisfies a skein relation; detects chirality and some knot properties
  • Calculated by resolving crossings, computing bracket polynomials for resulting links, and applying writhe correction
• HOMFLY-PT polynomial: Generalization of Alexander and Jones polynomials; satisfies a skein relation and encodes more knot information
  • Calculated using a state sum approach similar to the Kauffman bracket, with additional variables and relations
• Knot signature: Derived from the knot's Seifert matrix; measures the balance of positive and negative crossings in a knot diagram
• Knot genus: Minimum genus of any Seifert surface for the knot; relates to the knot's complexity and can be estimated using knot polynomials

Applications of Knot Equivalence and Invariants

• Topology and geometry: Knot theory is a branch of topology with connections to geometric structures like hyperbolic geometry
• Biology: Knots appear in DNA and proteins; knot invariants help understand their structure and function
  • Knot theory is used to study enzyme action on circular DNA and the supercoiling of DNA
• Chemistry: Knots are found in molecular structures like synthetic organic compounds and polymers; invariants aid in their classification and synthesis
• Physics: Knot theory is applied in quantum field theory, statistical mechanics, and the study of vortices in fluid dynamics
  • Knot invariants are used to classify and understand the behavior of these physical systems
• Computer science: Knot diagrams and invariants have applications in graph theory, coding theory, and cryptography
• Art and design: Knots have aesthetic and practical applications in fields like textile art, jewelry making, and decorative knotting

Problem-Solving Techniques

• Simplifying knot diagrams using Reidemeister moves to identify knot types and calculate invariants more easily
• Applying skein relations to compute knot polynomials recursively by resolving crossings and relating simpler knots or links
• Utilizing the properties of specific invariants to rule out potential equivalences or distinguish between knots
  • For example, using tricolorability to quickly identify non-equivalent knots or the Jones polynomial to detect chirality
• Constructing knot diagrams with desired properties or invariant values to prove statements or find counterexamples
• Employing algebraic and combinatorial techniques to derive and manipulate knot invariants, such as matrix calculations for the Alexander polynomial
• Collaborating with computational tools to calculate invariants for complex knots and explore knot databases
• Seeking connections between different invariants and their geometric or topological interpretations to gain insights into knot properties

Advanced Topics and Current Research

• Virtual knot theory: Studies knots in thickened surfaces and their diagrams with virtual crossings; extends classical knot theory
• Khovanov homology: Categorifies the Jones polynomial, assigning a bigraded homology theory to each knot; has applications in low-dimensional topology
• Knot concordance: Studies the equivalence of knots up to smooth embeddings in four-dimensional space; relates to knot surgery and slice knots
• Knot energies: Assign energy values to knots based on their geometric or physical properties; used in knot classification and applications
• Hyperbolic knot theory: Investigates the hyperbolic geometry of knot complements and its connections to knot invariants and structures
• Knots in higher dimensions: Generalizes knot theory to study embeddings of higher-dimensional spheres in higher-dimensional spaces
• Topological quantum computing: Uses braiding and fusion of anyons, modeled by knots and links, to perform quantum computations
• Machine learning in knot theory: Applies machine learning techniques to classify knots, predict invariants, and discover new patterns in knot data