Braids are fascinating mathematical objects that intertwine strands in 3D space. They form a group structure, allowing us to combine and manipulate them in unique ways. This concept bridges geometry and algebra, offering a visual and intuitive approach to abstract mathematical ideas.

Understanding braids and braid groups is crucial for grasping more advanced topics in knot theory. These structures provide powerful tools for analyzing knots and links, and their applications extend to various fields, including physics and cryptography.

Introduction to Braids and the Braid Group

Definition of braids and braid groups

• A braid on $n$ strands consists of $n$ non-intersecting paths in 3D space connecting $n$ points on a horizontal plane to $n$ points directly below on another horizontal plane
  • Paths may intertwine but never turn back upward (monotonic in the vertical direction)
  • Braids are equivalent if they can be continuously deformed into one another without intersecting the paths (ambient isotopy)
• The braid group on $n$ strands, $B_n$, is the set of all equivalence classes of braids on $n$ strands with the composition operation
  • Composition of braids is performed by stacking one braid on top of the other and connecting the endpoints (vertical stacking)
  • Identity element is the braid with $n$ straight vertical strands (trivial braid)
  • Every braid has an inverse obtained by reflecting the braid across a horizontal plane (mirror image)

Representation of braids

• Braid diagrams are 2D representations of braids viewed from the top
  • Each crossing represents a strand passing over or under another strand (over/under information)
  • Crossings are labeled with numbers to indicate the order in which they occur (crossing labels)
• Braid words are algebraic representations of braids using generators and their inverses
  • Generator $\sigma_i$ represents a crossing where the $i$-th strand passes over the $(i+1)$-th strand (positive crossing)
  • Inverse of a generator, $\sigma_i^{-1}$, represents a crossing where the $i$-th strand passes under the $(i+1)$-th strand (negative crossing)
  • A braid word is a sequence of generators and their inverses read from left to right (product of generators)
  • Example: the braid word $\sigma_1 \sigma_2^{-1} \sigma_1$ represents a braid with three crossings (2 positive, 1 negative)

Operations on braids

• Composition of braids is performed by stacking one braid diagram on top of the other and connecting the endpoints
  • For braid words, composition is achieved by concatenating the words (juxtaposition)
  • Example: if $\alpha = \sigma_1 \sigma_2$ and $\beta = \sigma_2^{-1} \sigma_1$, then $\alpha \beta = \sigma_1 \sigma_2 \sigma_2^{-1} \sigma_1$
• The inverse of a braid is obtained by reflecting the braid diagram across a horizontal line and reversing the direction of each crossing
  • For braid words, the inverse is obtained by reversing the order of the generators and replacing each generator with its inverse (reverse order and invert generators)
  • Example: if $\alpha = \sigma_1 \sigma_2^{-1} \sigma_1$, then $\alpha^{-1} = \sigma_1^{-1} \sigma_2 \sigma_1^{-1}$

Properties of braid groups

• The braid group $B_n$ is a group under the composition operation
  • Composition of braids is associative: $(\alpha \beta) \gamma = \alpha (\beta \gamma)$ for all braids $\alpha, \beta, \gamma$ (associativity)
  • Identity element is the braid with $n$ straight vertical strands (identity)
  • Every braid has an inverse as described in the previous section (inverses)
• The braid group is non-commutative for $n \geq 3$
  • Shown by considering the generators $\sigma_1$ and $\sigma_2$
  • $\sigma_1 \sigma_2 \neq \sigma_2 \sigma_1$ as the resulting braids are not equivalent (non-commutativity)
• The braid group $B_n$ is generated by the elements $\sigma_1, \sigma_2, \ldots, \sigma_{n-1}$
  • These generators satisfy the braid relations:
    1. $\sigma_i \sigma_j = \sigma_j \sigma_i$ for $|i - j| > 1$ (far commutativity)
    2. $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ for $1 \leq i \leq n-2$ (braid relation)
  • Any braid in $B_n$ can be expressed as a product of these generators and their inverses (generated group)