In these visualizations, time flows from top to bottom. The strands can be understood as the worldlines of particles moving forward in time. The Braid Group on N strands, denoted $B_N$, is the algebraic structure describing how N strands can be braided, defined by generators $\sigma_i$ where the $i$-th strand crosses over the $(i+1)$-th strand.
The generator $\sigma_i$ represents the elementary braid where the $i$-th strand crosses over the $(i+1)$-th strand. Its inverse, $\sigma_i^{-1}$, is the crossing of the $i$-th strand under the $(i+1)$-th. Unlike a free group, the generators of the Braid Group are constrained by relations, most notably the Braid Relation (or Yang-Baxter Equation): $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$.
Two knots are considered topologically equivalent if one can be deformed into the other through a series of three fundamental Reidemeister moves.
Adding or removing a twist in a single strand. $$ \sigma_i \sigma_i^{-1} = 1 $$
Sliding one strand over or under another. $$ \sigma_i \sigma_{i+1} \sigma_i^{-1} \sigma_{i+1}^{-1} = 1 $$
Sliding a strand across a crossing. $$ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} $$
The algebraic relation for the Type III move is known in physics as the Yang-Baxter equation. This equation is a fundamental condition for the integrability of models in statistical mechanics and quantum field theories, ensuring that scattering processes are consistent and allowing for exact solutions.