The Math of Knots – Alexander Polynomial

Knot theory classifies closed loops in 3D space. The trivial knot (unknot) can be untangled. The trefoil knot (three crossings) is non‑trivial. The Alexander polynomial Δ(t) is a knot invariant; for the trefoil, Δ(t) = t – 1 + t⁻¹. For the figure‑eight knot, Δ(t) = –t + 3 – t⁻¹. Invariants tell knots apart. The Jones polynomial (1984, Vaughan Jones) is more powerful. Knots appear in DNA recombination, quantum physics (anyons), and the study of the 3‑sphere. Understanding knots helps understand the structure of space itself.