1.2 Basic definitions: knots, links, and embeddings
2 min read•july 22, 2024
Knots and links are fundamental concepts in 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
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- 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 , , and (a simple loop)
Knots vs links
- A 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 (2 components), (3 components), and (2 components)
Embeddings in knot theory
- An 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 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 (, Jones polynomial) can distinguish between different knots based on their projections
Key Terms to Review (14)
Ambient space: Ambient space is the mathematical setting or environment in which geometric objects, such as knots or links, are situated. It provides the necessary framework to study the properties and relationships of these objects, often represented in three-dimensional Euclidean space. Understanding ambient space is crucial for discussing concepts like embeddings, as it dictates how these objects can be manipulated and analyzed within their given surroundings.
Borromean Rings: Borromean rings are a set of three linked circles in which no two circles are directly linked; removing any one ring causes the other two to become unlinked. This unique configuration illustrates important concepts in knot theory and serves as a classic example of multi-component links, showcasing how links can exist in a complex relationship without being interdependent.
Crossing Number: The crossing number of a knot or link is the minimum number of crossings in any diagram that represents it. This concept is fundamental as it helps in understanding the complexity of knots and links, providing a way to classify them and measure their intricacy through various representations.
Embedding: In knot theory, embedding refers to the representation of a knot or link as a continuous mapping from a one-dimensional space (such as a circle or line) into a three-dimensional space without intersections, thus preserving its structure. This concept is crucial as it allows mathematicians to study the properties of knots and links while maintaining their spatial relationships and topological features. Understanding embeddings also plays an important role in various applications, particularly in theoretical physics and string theory.
Figure-eight knot: The figure-eight knot is a type of knot commonly used in climbing, sailing, and rescue operations. It is known for its simplicity and reliability, providing a secure loop at the end of a rope, and it plays an essential role in understanding various aspects of knot theory.
Hopf Link: The Hopf link is a classic example of a two-component link in knot theory that consists of two circles that are linked together in a specific way, where each circle winds around the other. This link serves as a foundational example for understanding more complex links and their properties, while also playing a significant role in various polynomial invariants and fundamental group studies.
Knot: A knot is a closed loop formed by intertwining a strand of material, such as rope or string, in such a way that it cannot be undone without cutting the material. Knots are fundamental objects in the study of topology and knot theory, where they are used to understand more complex structures like links and embeddings. They can also be visualized within three-dimensional spaces and are important for exploring the characteristics and properties of 3-manifolds.
Knot invariant: A knot invariant is a property of a knot or link that remains unchanged under various transformations, specifically those that do not cut the knot or link. These invariants are crucial for distinguishing different knots and links from each other, allowing mathematicians to determine whether two knots are equivalent or not.
Knot Projection: Knot projection refers to a way of representing a knot or link by projecting it onto a plane, creating a two-dimensional diagram that captures the essential features of the knot's structure. This technique is crucial as it helps visualize knots, making them easier to analyze and manipulate, particularly when studying knot diagrams, transformations, and properties.
Link: In knot theory, a link is a collection of two or more closed curves in three-dimensional space that can be intertwined with each other but do not intersect. Unlike knots, which are single closed loops, links consist of multiple components that can have complex interrelations, leading to different types of links based on their arrangement and crossings. Understanding links helps in exploring the relationships and properties of multiple knots simultaneously.
Tangle: A tangle is a configuration of strands that can represent a knot or link, where the strands may cross over or under each other. Tangling is crucial for understanding how knots and links are formed, as it helps visualize and manipulate the relationships between strands. Tangles can be classified and analyzed to determine properties such as equivalence or how they can be transformed into one another without cutting the strands.
Trefoil Knot: A trefoil knot is the simplest nontrivial knot, resembling a three-looped configuration. It serves as a fundamental example in knot theory, illustrating key concepts such as knot diagrams, crossing numbers, and polynomial invariants, while also appearing in various applications across mathematics and science.
Unknot: An unknot is a simple loop in three-dimensional space that can be continuously transformed into a circle without any crossings or entanglements. It serves as the fundamental building block in knot theory, distinguishing between knots and links and laying the groundwork for understanding how various embeddings behave.
Whitehead link: The Whitehead link is a specific example of a 2-component link that consists of two intertwined loops which cannot be separated without cutting one of the loops. This link is notable for its unique properties and serves as an important example in knot theory, particularly illustrating concepts of links, embeddings, and invariants associated with multi-component structures.