Algebraic topology explores fundamental structures like knots, links, and braids through mathematical frameworks. A knot is defined as a simple closed curve in three-dimensional space, where two knots are equivalent if one can be continuously deformed into the other. The field distinguishes between tame embeddings (standard knots) and wild embeddings, which exhibit more complex topological properties.

Key concepts include Reidemeister moves, which provide a complete set of transformations for determining when two knot diagrams represent the same knot. The theory also encompasses Seifert surfaces, which are orientable surfaces bounded by knots, and allow for defining knot invariants like genus. Prime knots serve as the building blocks of knot theory, analogous to prime numbers in arithmetic.

Advanced topics include the Jones polynomial, a powerful invariant that can distinguish many inequivalent knots, and braid groups, which provide algebraic structures for studying braids on multiple strands. The fundamental group of a knot's complement serves as a basic invariant, though it cannot always distinguish between different knots. These mathematical tools have applications in fields ranging from DNA analysis to quantum computing, demonstrating the practical relevance of abstract topological concepts.