11.1 Multi-component links and their properties
2 min read•july 22, 2024
Multi-component links expand our understanding beyond single knots. They involve multiple intertwined curves, like the and . Each component can have its own orientation, and links can be classified as splittable or non-splittable.
and are key concepts in analyzing multi-component links. These ideas help us determine when two links are essentially the same, sharing properties like component count, , and individual knot types.
Multi-Component Links
Types of multi-component links
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- Hopf link consists of two linked circles that cannot be separated without cutting one of them
- Borromean rings are three linked circles where removing any one of them leaves the other two unlinked
- has two linked circles that are not splittable but become splittable after removing one full twist from either component
- Other examples include the , , and the
Components and orientations of links
- Identify the components of a link by visually inspecting the and locating each separate simple closed curve
- Each separate curve in the diagram represents one component of the link
- Determine the orientation of a component by assigning a direction (clockwise or counterclockwise) to each component
- Indicate orientations using arrows on the link diagram
- The choice of orientation for each component is arbitrary but must remain consistent throughout the analysis of the link
Link equivalence and ambient isotopy
- Link equivalence is a relation between two links that can be transformed into each other through continuous deformations without cutting or passing through itself
- Ambient isotopy refers to continuously deforming the link in 3-dimensional space
- Equivalent links are related by an ambient isotopy and share the same topological properties
- These properties include the number and type of components, the linking number between components, and the knot type of each component
Splittable vs non-splittable links
- A link is splittable if its components can be separated into two or more subsets without cutting or passing through any component
- In a , a 2-sphere embedded in or separates the components into two or more subsets
- The subsets of components in a splittable link are not linked with each other (Hopf link is an example of a splittable link)
- A link is non-splittable if its components cannot be separated into subsets without cutting or passing through any component
- In a , no 2-sphere can separate the components into subsets
- Non-splittable links have components that are inherently linked with each other (Borromean rings and Whitehead link are examples of non-splittable links)
- The splittability of a link is a topological property that remains invariant under ambient isotopy
Key Terms to Review (20)
2-component link: A 2-component link is a specific type of link in knot theory that consists of two distinct loops or components that are interlinked with each other. The important feature of a 2-component link is that while the two loops cannot be separated without cutting them, each loop remains an independent entity, allowing for various properties and operations to be examined, such as linking number and knot invariants.
3-component link: A 3-component link is a specific type of link in knot theory that consists of three distinct loops or components that are interlinked with each other. Each component is a simple closed curve, and their arrangement can lead to various properties and classifications of the link, including whether it can be untangled without cutting any of the components. Understanding 3-component links involves exploring their connectivity and how they relate to different link invariants.
Ambient isotopy: Ambient isotopy refers to a continuous deformation of a space in which a knot or link can be transformed into another without cutting the strands, allowing the surrounding space to change shape while keeping the knot itself intact. This concept is crucial for determining when two knots are considered equivalent, as it focuses on the relationship between the knot and its environment.
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.
Brunnian link: A Brunnian link is a specific type of multi-component link where removing any single component from the link causes the entire link to become disconnected. This property makes Brunnian links interesting in knot theory, as they showcase unique behaviors distinct from other types of links. These links highlight important concepts such as unlinking and the relationships between components within a multi-component system.
Charles Livingston: Charles Livingston is a prominent figure in knot theory known for his contributions to the study of ambient isotopy and knot equivalence. His work has been instrumental in understanding how knots can be manipulated without cutting the string, focusing on the conditions under which different knots are considered equivalent. Through his research, he has provided valuable insights into multi-component links and their unique properties, enhancing the overall knowledge of knot theory.
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.
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.
John Conway: John Conway was a prominent British mathematician known for his groundbreaking contributions to various areas of mathematics, particularly in knot theory and the study of braids. His work includes the introduction of the notation for knots, which has become essential for understanding knot equivalence and classification. Conway's influence extends to multi-component links, enriching the study of topology and its applications in different mathematical domains.
Link diagram: A link diagram is a visual representation of a link, consisting of a collection of closed curves, which are often depicted in two-dimensional space. Each closed curve represents a component of the link, and the intersections or crossings between these curves indicate how the components interlace with each other. Understanding link diagrams is crucial for analyzing properties like isotopy, as well as studying multi-component links and their relationships to algebraic structures such as Khovanov homology.
Link equivalence: Link equivalence is a relation between two links that signifies they can be transformed into each other through a series of allowable manipulations, such as twisting or moving components without cutting the strands. This concept is crucial for understanding how multi-component links behave and interact, as well as for determining their properties through numerical invariants like the linking number.
Linking number: The linking number is an integer that represents the degree of linkage between two or more components of a link in knot theory. It indicates how many times the strands of one component wrap around the strands of another component, which helps in distinguishing different types of links and understanding their topological properties.
Negative Crossing: A negative crossing is a specific type of crossing in knot diagrams where a strand of the knot goes under another strand when two strands intersect. This crossing is important in knot theory as it helps to distinguish between different knots and links. The way strands cross over or under each other plays a crucial role in defining the structure of a knot, influencing calculations related to knot invariants and properties of multi-component links.
Non-splittable link: A non-splittable link is a type of link in knot theory that cannot be separated into two or more components without cutting. This property indicates that all components of the link are interconnected in such a way that it is impossible to isolate any single component from the others without breaking the link. Non-splittable links exhibit important characteristics regarding their topology and interactions among their components, which plays a crucial role in understanding multi-component links and their properties.
Positive Crossing: A positive crossing occurs in a knot diagram when a strand of the knot crosses over another strand, creating a right-handed twist. This type of crossing is important for understanding the structure of knots and links, as it helps to determine their topological properties. The distinction between positive and negative crossings can influence various calculations and properties associated with knots, including their Alexander polynomial and behaviors in multi-component links.
Prime link: A prime link is a multi-component link that cannot be represented as the connected sum of two non-trivial links. This means that it cannot be decomposed into simpler links and is considered an 'atomic' or 'building block' of link theory. Recognizing prime links is essential for understanding the structure and classification of more complex links, as they serve as the fundamental units from which all links are constructed.
Solomon's Knot: Solomon's Knot is a specific type of multi-component link that consists of two interlocked loops that can be represented as a square knot. This knot is significant in knot theory as it showcases the properties and behaviors of multi-component links, particularly in how they can be manipulated and understood through various mathematical approaches.
Splittable Link: A splittable link is a type of link in knot theory that can be separated into two or more components without cutting any of the strands. This property highlights the distinction between multi-component links and non-splittable links, emphasizing the topological behavior of the strands. Understanding splittable links is crucial for exploring the properties of multi-component links, as it leads to insights about their structure, classification, and the manipulation of knots within the broader context of knot theory.
Star of David link: The Star of David link is a specific multi-component link that consists of two intertwined triangles, resembling the Star of David symbol. This link is notable in knot theory as it represents a nontrivial example of a two-component link, showcasing unique properties such as its linking number and the ability to be represented in various projections. Understanding this link helps in studying how multiple components can interact and the implications of their arrangement in knot theory.
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.