Knot Diagrams

A knot is an embedding of a circle S¹ into 3-space, studied up to ambient isotopy. The crossing number is the minimum crossings in any diagram. Writhe counts signed crossings (each +1 or -1 by right/left-hand rule) — not a knot invariant but important for framing. Reidemeister's theorem: two diagrams represent the same knot iff connected by a sequence of three local moves (R1: curl, R2: push-off, R3: triangle slide).