🪢Knot Theory Unit 9 – Knot Tabulation and Classification

Introduction to Knot Tabulation

• Knot tabulation involves systematically organizing and cataloging different types of knots based on their properties and characteristics
• Enables researchers to study knots in a structured manner and identify relationships between different knot types
• Facilitates the discovery of new knot invariants and the development of classification schemes
• Helps in understanding the complexity of knots and their behavior under various transformations
• Provides a foundation for applications in fields such as biology (DNA topology), physics (quantum field theory), and chemistry (molecular structures)

Basic Concepts and Terminology

• A knot is a closed, non-self-intersecting curve embedded in three-dimensional space
  • Can be thought of as a loop of rope with its ends joined together
• A link consists of multiple knots that are intertwined or linked together
• The crossing number of a knot is the minimum number of crossings in any diagram representing the knot
• A knot is said to be prime if it cannot be decomposed into simpler knots by cutting and reconnecting the strands
• The unknot, also known as the trivial knot, is the simplest possible knot with no crossings
• Knot equivalence refers to the notion that two knots are considered the same if they can be continuously deformed into each other without cutting or passing through itself
• Chirality describes whether a knot is distinct from its mirror image
  • A knot is chiral if it is not equivalent to its mirror image, and achiral otherwise

Knot Diagrams and Projections

• A knot diagram is a two-dimensional representation of a knot obtained by projecting the knot onto a plane
• In a knot diagram, crossings are represented by breaks in the lines, with one strand passing over the other
• The over/under information at each crossing is crucial for distinguishing different knots
• A knot projection is a knot diagram without the over/under information at the crossings
• Alternating knots are those whose diagrams have alternating over and under crossings as one travels along the knot
• Reduced diagrams are those with the minimum number of crossings possible for a given knot
• Different diagrams can represent the same knot, as long as they can be transformed into each other using Reidemeister moves

Reidemeister Moves

• Reidemeister moves are local transformations that can be applied to a knot diagram without changing the underlying knot type
• There are three types of Reidemeister moves:
  • Type I: A twist can be added or removed from a strand
  • Type II: Two strands can be moved to create or remove a pair of crossings
  • Type III: A strand can be moved across a crossing, changing the over/under relationship
• Reidemeister's theorem states that any two diagrams representing the same knot can be transformed into each other using a sequence of Reidemeister moves
• Reidemeister moves are essential for proving knot equivalence and for simplifying knot diagrams
• The number of Reidemeister moves required to transform one diagram into another can be used as a measure of knot complexity

Knot Invariants

• Knot invariants are properties of knots that remain unchanged under continuous deformations, including Reidemeister moves
• Invariants are used to distinguish between different knot types and to classify knots
• The crossing number is a simple invariant that counts the minimum number of crossings in any diagram of a knot
• The knot group is an algebraic invariant that encodes information about the fundamental group of the knot complement
• The Alexander polynomial is a polynomial invariant that can be computed from a knot diagram and provides information about the knot's structure
• The Jones polynomial is another polynomial invariant that is sensitive to the chirality of knots and can detect certain properties not captured by the Alexander polynomial
• Knot invariants can be used to prove that two knots are distinct, but the converse is not always true (i.e., two knots with the same invariants may still be different)

Tabulation Methods

• Knot tabulation involves creating a systematic list of all possible knots up to a certain crossing number
• The tabulation process typically starts with the unknot and progressively adds crossings to generate more complex knots
• Knots are often organized by their crossing number, with separate tables for prime and composite knots
• The Rolfsen table is a well-known knot table that lists all prime knots up to 10 crossings and provides diagrams for each knot
• The Hoste-Thistlethwaite table extends the tabulation to 16 crossings and includes both prime and composite knots
• Tabulation methods often rely on computational algorithms to generate and classify knots efficiently
• Challenges in knot tabulation include dealing with the rapid growth of the number of knots as the crossing number increases and ensuring that all distinct knots are accounted for

Classification Techniques

• Knot classification aims to organize knots into families or classes based on their properties and relationships
• The most basic classification is based on the crossing number, which groups knots by their minimal diagram complexity
• Knots can also be classified by their chirality, with chiral knots forming distinct classes from achiral knots
• The genus of a knot, which measures the complexity of the surface obtained by seifing the knot, provides another classification criterion
• Hyperbolic knots, which have a hyperbolic geometry in their complement, form a significant class of knots with distinct properties
• Satellite knots are constructed by embedding one knot (the companion) into the tubular neighborhood of another knot (the pattern), forming a hierarchical classification scheme
• Torus knots are a special class of knots that can be embedded on the surface of a torus without self-intersections
• Classification schemes based on knot invariants, such as polynomial invariants or knot homology theories, provide more refined ways to distinguish and organize knots

Applications and Further Study

• Knot theory has found applications in various scientific fields beyond pure mathematics
• In biology, knot theory is used to study the topology of DNA molecules and the mechanisms of DNA replication and recombination
• Knots appear in physics, particularly in the study of quantum field theories and the behavior of subatomic particles
• In chemistry, knot theory is applied to the analysis of molecular structures and the synthesis of complex molecules with knotted topologies
• Knot theory also has connections to other areas of mathematics, such as low-dimensional topology, algebraic topology, and representation theory
• Open problems in knot theory include the classification of knots in higher dimensions, the study of virtual knots and links, and the development of new invariants and computational methods
• The study of knot tabulation and classification continues to be an active area of research, with ongoing efforts to extend tables to higher crossing numbers and to explore new ways of organizing and understanding the rich structure of knots