Braid representations offer a powerful way to study knots. By transforming knots into braids and back, we gain new insights into their structure and properties. This technique opens up exciting avenues for computing knot invariants and polynomials.
tells us every knot can be represented as a closed braid. This allows us to apply braid theory tools to knot problems, leading to breakthroughs in understanding and developing new invariants.
Braid Representations of Knots
Knots as braid closures
Top images from around the web for Knots as braid closures
File:Blue Figure-Eight Knot.png - Wikimedia Commons View original
Is this image relevant?
gt.geometric topology - How to see isometries of figure 8 knot complement - MathOverflow View original
Is this image relevant?
File:Blue Figure-Eight Knot.png - Wikimedia Commons View original
Is this image relevant?
gt.geometric topology - How to see isometries of figure 8 knot complement - MathOverflow View original
Is this image relevant?
1 of 2
Top images from around the web for Knots as braid closures
File:Blue Figure-Eight Knot.png - Wikimedia Commons View original
Is this image relevant?
gt.geometric topology - How to see isometries of figure 8 knot complement - MathOverflow View original
Is this image relevant?
File:Blue Figure-Eight Knot.png - Wikimedia Commons View original
Is this image relevant?
gt.geometric topology - How to see isometries of figure 8 knot complement - MathOverflow View original
Is this image relevant?
1 of 2
Alexander's theorem establishes every knot or link represents the
Braid consists of strings running top to bottom with finitely many crossings between adjacent strings
Closure of a braid obtained by connecting top and bottom endpoints of each string (, )
Representing a knot as a braid involves choosing a projection plane, selecting a perpendicular "vertical" direction, and manipulating the to consist of monotonic strands with crossings only between adjacent strands
(conjugation, stabilization/destabilization) applied to a braid yield equivalent braids representing the same knot
Braid representation via Alexander's theorem
Begin with a regular knot diagram
Select a basepoint and direction of travel on the knot diagram
Number the crossings sequentially while following the knot diagram in the chosen direction
Construct a braid by creating straight strands corresponding to regions between crossings
Each crossing in the knot diagram maps to a crossing between two adjacent strands in the braid
Over/under information at each crossing is preserved in the
Closing the resulting braid by connecting top and bottom endpoints represents the original knot (trefoil knot, )
Braid index vs knot complexity
of a knot is the minimum number of strands needed to represent the knot as a closed braid
Braid index provides a lower bound on the (minimum crossings in any knot diagram)
Higher crossing numbers generally correspond to higher braid indices, but knots with the same crossing number can have different braid indices (5₁ and 5₂ knots)
Braid index is not a complete ; non-equivalent knots can have the same braid index (6₁ and 6₂ knots)
No general algorithm exists for determining the minimum braid index of a given knot
Applications of braid theory
Braid representations enable computation of knot invariants and polynomials
defined using , interpreted in terms of braids
Kauffman bracket of a braid closure is a weighted sum over all possible smoothings of the crossings
Jones polynomial obtained from Kauffman bracket by normalizing and accounting for writhe of the knot diagram
computed using skein relation applied to braid representations, generalizing Jones and Alexander polynomials
, a categorification of the Jones polynomial, studied using braid representations and a cube of resolutions construction
Knot invariants defined using representations of the (, )
Key Terms to Review (19)
Alexander's Theorem: Alexander's Theorem states that every knot in three-dimensional space can be represented as a finite union of non-intersecting arcs, or 'tangles', with endpoints on a circle. This theorem bridges the understanding of knots through ambient isotopy and equivalence by providing a foundation for exploring how knots can be manipulated and classified based on their structure and properties.
Braid group: The braid group is a mathematical structure that consists of equivalence classes of braids, which are defined by the ways in which strands can be intertwined without cutting or moving the endpoints. This concept connects deeply with knot theory, particularly in how braids can represent knots and how they relate to various invariants and mathematical transformations.
Braid index: The braid index is the minimum number of strands required to represent a braid. It serves as a fundamental property that connects braids to knots, as each knot can be represented by a braid with a specific braid index. This concept is essential for understanding how different knots can arise from braids and plays a crucial role in the study of their relationships.
Braid representation: Braid representation refers to a mathematical way of depicting braids using strands that intertwine in space, often represented visually through a series of crossings. This concept is crucial in understanding the relationships between different topological entities and their transformations, especially when considering isotopies and the connections between braids and knots. It also plays a significant role in Artin's braid theory, which formalizes the study of braids and their properties.
Burau Representation: The Burau representation is a mathematical concept used in knot theory, which associates each braid with a matrix representation. It captures the relationships between braids and knots by providing a way to analyze the properties of braids through linear algebra. This representation plays a significant role in understanding how braids can be transformed into knots and helps to establish connections between the two fields.
Cinquefoil knot: The cinquefoil knot is a type of mathematical knot that is recognized as the simplest nontrivial knot with 5 crossings. It serves as an important example in the study of knot theory, particularly in understanding the properties of knot groups, analyzing behaviors in polymer physics, and exploring the connections between braids and knots.
Closure of a braid: The closure of a braid is a process where the endpoints of a braid are joined together to form a loop, effectively creating a knot. This transformation connects braids to knots, illustrating how braids can serve as a method for constructing various types of knots, depending on how the strands are intertwined and closed. Understanding the closure of a braid helps in exploring the intricate relationships between different knot types and their representations through braiding techniques.
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.
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.
Homfly-pt polynomial: The homfly-pt polynomial is a powerful invariant in knot theory that generalizes both the Alexander and Jones polynomials. It provides a way to distinguish knots and links based on their topology, and is expressed as a two-variable polynomial that captures more information than its predecessors, making it useful in various mathematical contexts.
Jones Polynomial: The Jones polynomial is a significant knot invariant that assigns to each oriented knot or link a polynomial in one variable, often denoted as $V(t)$. It captures essential information about the knot's topology and is derived using a particular method involving knot diagrams and the Kauffman bracket, providing a deeper understanding of knot theory.
Kauffman Bracket: The Kauffman bracket is a polynomial invariant of knots and links in three-dimensional space, defined through a recursive formula that involves the crossings of a knot diagram. It serves as a foundational tool in knot theory, connecting to other key concepts like the Kauffman polynomial and the Jones polynomial while providing insights into the relationships between braids and knots.
Khovanov Homology: Khovanov homology is a powerful invariant in knot theory that categorifies the Jones polynomial, providing a richer structure to study knots and links. It connects the world of homology theories to knot invariants and enhances our understanding of their relationships, particularly through its ability to distinguish knots that are not identifiable through traditional means. The construction of Khovanov homology also reveals interesting connections between braids and knots, highlighting the interplay between these concepts in knot theory.
Knot complexity: Knot complexity refers to the measurement of how complicated a knot is, often assessed by the minimum number of crossings or the simplest braid that can represent it. This concept connects knots and braids by demonstrating how different braids can produce knots with varying degrees of complexity. Understanding knot complexity helps in classifying knots and determining their properties in relation to their braid representations.
Knot diagram: A knot diagram is a two-dimensional representation of a knot or link, typically drawn on a plane to show its crossings, over- and under-relationships between strands, and orientation. This visual representation helps in analyzing various properties of knots, such as their chirality and isotopy, and serves as a foundational tool for further studies in knot theory.
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.
Lawrence-Krammer Representation: The Lawrence-Krammer representation is a mathematical framework that provides a way to represent braids as matrices, allowing for the study of their properties in relation to knots. This representation links braids to their associated knot types through a series of matrix equations, which can then be analyzed using algebraic methods. It plays a significant role in understanding how different braid structures can lead to specific knot formations and their invariants.
Markov Moves: Markov moves are a set of operations used to manipulate braids and knots in a way that preserves their equivalence. These moves are crucial for understanding the relationship between braids and knots, as they allow one to transform a braid into another while maintaining its topological properties. By applying Markov moves, one can demonstrate that two braids represent the same knot or link.
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.