knot theory unit 10 study guides

Definition and Basics

• Braids are a mathematical object consisting of a set of strings that cross over and under each other in a specific pattern
• The strings in a braid are assumed to be attached at the top and bottom to two parallel planes
• Braids can be thought of as a way to represent the intertwining of strings or strands in three-dimensional space
• The number of strings in a braid is called the braid index and is typically denoted by the variable $n$
• Two braids are considered equivalent if one can be continuously deformed into the other without cutting or joining any strings
• The set of all braids with a fixed number of strings forms a group under the operation of braid composition
  • Braid composition involves stacking one braid on top of another and connecting the corresponding strings
• The identity element in the braid group is the braid where all strings go straight down without any crossings

Geometric Representation

• Braids can be represented geometrically as a set of strings embedded in three-dimensional space
• Each string in the braid starts at a point on the top plane and ends at a corresponding point on the bottom plane
• The strings are allowed to cross over and under each other, but they cannot intersect or pass through each other
• The geometric representation of a braid captures the essential features of the braid, such as the number of strings and the pattern of crossings
• Braids can be manipulated geometrically by performing a sequence of braid moves, which are local deformations that change the crossing pattern
  • Examples of braid moves include the Reidemeister moves and the braid group generators
• The geometric representation of braids is closely related to the study of knots and links
  • Closing a braid by connecting the corresponding top and bottom endpoints results in a knot or link

Algebraic Structure

• The braid group, denoted by $B_n$, is the group formed by the set of all braids with $n$ strings under the operation of braid composition
• The braid group has a presentation in terms of generators and relations
  • The generators of the braid group are the elementary braids $\sigma_i$, where $i = 1, 2, \ldots, n-1$
  • Each generator $\sigma_i$ represents a crossing between the $i$-th and $(i+1)$-th strings, with the $i$-th string passing over the $(i+1)$-th string
• The braid group satisfies the following relations:
  • $\sigma_i \sigma_j = \sigma_j \sigma_i$ for $|i-j| \geq 2$ (far commutativity)
  • $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ for $i = 1, 2, \ldots, n-2$ (braid relation)
• The braid group is an infinite, non-abelian group for $n \geq 3$
• The braid group has a natural homomorphism onto the symmetric group $S_n$, obtained by forgetting the crossing information and considering only the permutation of strings

Braid Group Properties

• The braid group $B_n$ is a torsion-free group, meaning that no element of the group has finite order except for the identity
• The braid group has a solvable word problem, which means there is an algorithm to determine whether two braid words represent the same braid
• The braid group has a faithful representation in the automorphism group of a free group
  • This representation, known as the Artin representation, allows for the study of braids using algebraic and combinatorial techniques
• The braid group is a Garside group, which means it has a special element called the Garside element and a normal form for its elements
  • The Garside structure provides a way to solve the word problem and to study the algebraic properties of the braid group
• The braid group has connections to various areas of mathematics, including topology, algebra, and geometry
• The pure braid group, denoted by $P_n$, is the kernel of the homomorphism from the braid group to the symmetric group
  • Elements of the pure braid group are braids where each string ends at the same position as it started

Braid Diagrams and Notation

• Braid diagrams are two-dimensional representations of braids that capture the essential crossing information
• In a braid diagram, the strings are represented by vertical lines, and the crossings are indicated by breaks in the lines
  • A positive crossing (overcrossing) is denoted by a break in the lower string
  • A negative crossing (undercrossing) is denoted by a break in the upper string
• Braid diagrams are read from top to bottom, with the convention that the string on the left passes over the string on the right at each crossing
• Braid words are a notation for representing braids algebraically using the generators $\sigma_i$ and their inverses $\sigma_i^{-1}$
  • A positive crossing between the $i$-th and $(i+1)$-th strings is represented by $\sigma_i$
  • A negative crossing between the $i$-th and $(i+1)$-th strings is represented by $\sigma_i^{-1}$
• Braid words are read from left to right, with the convention that the generators are applied from top to bottom
• The closure of a braid is obtained by connecting the corresponding top and bottom endpoints of the strings in the braid diagram
  • The closure of a braid results in a knot or link diagram

Relation to Knots

• Braids and knots are closely related mathematical objects in the field of knot theory
• Every knot or link can be represented as the closure of some braid
  • This is known as Alexander's theorem and provides a fundamental connection between braids and knots
• The braid index of a knot is the minimum number of strings required to represent the knot as the closure of a braid
  • Determining the braid index of a knot is an important problem in knot theory and has applications in various fields
• Markov's theorem characterizes when two braids have equivalent closures as knots or links
  • Markov moves, which include conjugation and stabilization, are used to manipulate braids while preserving the knot type of their closures
• The Jones polynomial, a knot invariant, can be computed using the representation theory of the braid group
  • The Markov trace on the Iwahori-Hecke algebra of the braid group gives rise to the Jones polynomial
• Studying knots via their braid representations has led to significant advances in knot theory and low-dimensional topology

Applications and Examples

• Braids have applications in various fields, including physics, chemistry, and biology
• In statistical mechanics, braids are used to model the entanglement of polymers and the behavior of anyons in two-dimensional systems
  • The braid group statistics of anyons have potential applications in topological quantum computation
• In molecular biology, braids are used to describe the topology of DNA and the action of certain enzymes on DNA strands
  • Enzymes such as recombinases and topoisomerases can perform operations on DNA that are analogous to braid moves
• Braids are also used in the study of fluid dynamics and the motion of vortices in fluids
  • The braiding of vortex lines in three-dimensional space can be described using braid theory
• In mathematics, braids have connections to various areas, including group theory, representation theory, and low-dimensional topology
  • The braid group is an important example of a non-commutative group and has been studied extensively in algebra and topology
• Braids have also found applications in computer science, particularly in the areas of cryptography and quantum computation
  • Braid group cryptography uses the complexity of the braid group to construct secure cryptographic protocols

Advanced Concepts

• The Yang-Baxter equation is a fundamental equation in mathematical physics that has close connections to braid theory
  • Solutions to the Yang-Baxter equation, known as R-matrices, give rise to representations of the braid group
• The Iwahori-Hecke algebra of the braid group is a deformation of the group algebra of the braid group
  • The Iwahori-Hecke algebra has important applications in representation theory and the construction of knot invariants
• The Birman-Murakami-Wenzl (BMW) algebra is another algebraic structure related to the braid group
  • The BMW algebra is a quotient of the braid group algebra and has connections to the Kauffman polynomial and other knot invariants
• The Garside normal form and the Dehornoy ordering provide a way to study the structure and properties of the braid group
  • The Garside normal form gives a unique representative for each element of the braid group and has applications in solving the word problem
• The braid group has a faithful action on the free group, known as the Artin action
  • This action has been used to study the algebraic and geometric properties of the braid group and its subgroups
• The braid group has connections to mapping class groups of surfaces and the theory of diffeomorphisms
  • The mapping class group of the punctured disk is isomorphic to the braid group, providing a geometric interpretation of braids