Knots and links are fundamental concepts in knot theory. Knots are closed curves in 3D space, while links are collections of intertwined knots. Understanding their definitions and differences is crucial for grasping the basics of this field.
Embeddings play a key role in knot theory, mapping objects into higher-dimensional spaces. Knot projections help visualize 3D knots in 2D, preserving crossing information. These concepts form the foundation for studying knot behavior and classification.
Knots and Links
Definition of knots
- A knot is a closed curve in 3D space without endpoints forming a continuous loop
- The curve must be non-self-intersecting or simple at any point
- Examples include the trefoil knot, figure-eight knot, and unknot (a simple loop)
Knots vs links
- A link is a collection of multiple intertwined or linked knots, each called a component
- Links are classified by the number of components they contain (2-component link, 3-component link, etc.)
- Examples of links include the Hopf link (2 components), Borromean rings (3 components), and Whitehead link (2 components)
Embeddings in knot theory
- An embedding maps an object into a higher-dimensional space without self-intersection
- In knot theory, an embedding maps a 1D curve into 3D space
- Knots and links are images of embeddings of circles into 3D space
- A knot is the image of an embedding of a single circle
- A link is the image of an embedding of multiple circles
- Studying embedding properties helps understand knot and link behavior and classification
Knots and their projections
- A knot projection is a 2D representation of a 3D knot obtained by projecting onto a plane
- The projection may contain crossings where the curve appears to pass over or under itself
- In a knot projection, over and under information at each crossing is preserved using breaks in the underpass to reconstruct the original 3D knot
- A knot projection may not accurately represent the 3D structure, as different projections of the same knot may appear visually distinct
- Knot invariants (crossing number, Jones polynomial) can distinguish between different knots based on their projections