🪢Knot Theory Unit 2 – Knot Diagrams and Reidemeister Moves

What's the Deal with Knot Diagrams?

• Knot diagrams are 2D representations of 3D knots that capture their essential structure and properties
• Consist of a closed loop with a finite number of crossings where the strand passes over or under itself
• Enable mathematicians to study knots abstractly without relying on physical models or embeddings in 3D space
• Provide a standardized way to compare and classify different knots based on their diagram representations
• Allow for the application of combinatorial and algebraic techniques to analyze knot invariants and properties
• Serve as the foundation for the development of knot polynomials (Alexander, Jones, HOMFLY) and other algebraic knot invariants
• Play a crucial role in understanding the connections between knots, links, braids, and other topological objects

Untangling the Basics: Key Concepts

• A knot is a closed loop embedded in 3D space that does not intersect itself and cannot be untangled into a simple circle without cutting
• Crossings occur when the strand of the knot passes over or under itself in the diagram representation
  • Overpasses and underpasses determine the local structure of the knot at each crossing
• Knot equivalence means two knots are considered the same if they can be continuously deformed into each other without cutting or passing through itself
• Knot invariants are properties or quantities associated with a knot that remain unchanged under continuous deformations (Reidemeister moves)
  • Examples include crossing number, unknotting number, and various knot polynomials
• The unknot is the simplest possible knot, equivalent to a simple circle with no crossings
• Links are generalizations of knots consisting of multiple intertwined closed loops
• Knot chirality refers to the handedness or orientation of a knot, with some knots having distinct left-handed and right-handed forms

Drawing the Line: How to Make Knot Diagrams

• Start by sketching a closed loop with the desired number of crossings, ensuring the strand only intersects itself transversely
• At each crossing, clearly indicate which strand passes over (overpass) and which passes under (underpass) using gaps or breaks in the lines
• Assign labels or orientations to the strands if needed for further analysis or computation of invariants
• Ensure the diagram is as simple and uncluttered as possible, minimizing unnecessary twists or bends
  • Aim for a minimal crossing number representation when possible
• Verify that the diagram accurately represents the intended knot and its key features
• Consider using graph paper, vector graphics software, or specialized knot theory tools for creating precise and easily manipulable diagrams
• Practice drawing common knots (trefoil, figure-eight, cinquefoil) and their mirror images to build familiarity with the diagramming process

The Reidemeister Moves: Knot Transformations 101

• Reidemeister moves are local diagram transformations that preserve knot equivalence, allowing for the simplification and standardization of knot diagrams
• There are three types of Reidemeister moves, each addressing a specific local configuration in the knot diagram
  • Type I (Twist): Adds or removes a single twist in the strand, introducing or eliminating a small loop
  • Type II (Poke): Adds or removes two crossings in a strand that passes over or under itself, like pulling a strand through a loop
  • Type III (Slide): Moves a strand over or under a crossing, sliding it past the intersection without changing the overall knot type
• Any two equivalent knots can be transformed into each other through a finite sequence of Reidemeister moves
• Reidemeister moves are reversible, meaning they can be applied in both directions to simplify or complicate a knot diagram
• The moves are named after Kurt Reidemeister, who introduced them in the 1920s as part of his foundational work in knot theory
• Understanding and applying Reidemeister moves is essential for proving knot invariants, classifying knots, and studying knot equivalence

Putting It All Together: Practice and Applications

• Practice drawing and manipulating knot diagrams for common knots (trefoil, figure-eight, cinquefoil) and their mirror images
• Apply Reidemeister moves to simplify and standardize knot diagrams, aiming for minimal crossing number representations
• Compute knot invariants (crossing number, unknotting number) for given knot diagrams to gain familiarity with their properties and behavior
• Explore the connections between knot diagrams, braids, and link diagrams, learning how to convert between these representations
• Study the applications of knot theory in various fields, such as:
  • Biology: Modeling DNA structure and replication, protein folding
  • Chemistry: Synthesizing molecular knots and links, studying topological isomers
  • Physics: Analyzing quantum entanglement, exploring topological quantum computation
• Engage with knot theory research by reading introductory papers, attending seminars, or participating in online forums and discussions

Mind-Bending Examples: Cool Knots to Try

• Trefoil knot: The simplest nontrivial knot, consisting of three crossings in a circular pattern
• Figure-eight knot: A non-invertible knot with four crossings, resembling a figure-eight shape
• Cinquefoil knot: A knot with five crossings arranged in a pentagonal pattern, also known as the Solomon's seal knot
• Borromean rings: A link of three interlocked circles, where removing any one circle frees the other two
  • Demonstrates the concept of Brunnian links, where removing any component unlinks the remaining components
• Torus knots: Knots that can be drawn on the surface of a torus without self-intersection, denoted by two integers (p, q) representing the number of times the knot winds around the torus
• Pretzel knots: Knots formed by intertwining a number of strands in a pattern resembling a pretzel, with each strand representing a sequence of half-twists
• Mobius strip: A non-orientable surface with only one side and one boundary component, formed by taking a strip of paper, giving it a half-twist, and joining the ends

Beyond the Basics: Advanced Topics and Connections

• Knot polynomials: Algebraic invariants associated with knots, including the Alexander polynomial, Jones polynomial, and HOMFLY polynomial
  • These polynomials encode information about the knot's structure and properties, allowing for finer distinctions between knots
• Knot tabulation and classification: The systematic enumeration and organization of knots by their crossing number and other properties
  • Knot tables provide a comprehensive catalog of knots up to a certain crossing number, facilitating the study and comparison of knot types
• Hyperbolic knots: Knots whose complement (the space around the knot) admits a complete hyperbolic metric, connecting knot theory with hyperbolic geometry
• Knot concordance: The study of knots up to concordance, a weaker equivalence relation than ambient isotopy
  • Two knots are concordant if they form the boundary of a smooth embedding of an annulus in four-dimensional space
• Khovanov homology: A powerful knot invariant that categorifies the Jones polynomial, providing a richer algebraic structure for studying knots and links
• Topological quantum field theories (TQFTs): Mathematical frameworks that associate algebraic objects to topological spaces and manifolds, with knots playing a central role in their construction and analysis
  • TQFTs have deep connections to quantum physics and provide a bridge between knot theory and other areas of mathematics, such as representation theory and category theory