Introduction to Knot Theory

Algebraic Topology · Axiom Academy

LESSON Introduction to Knot Theory Exploring knots as embeddings in 3-space and their topological invariants 1. Knots as Embeddings S¹ → S³ Intuitively, a knot is a closed loop in 3-dimensional space with no self-intersections. We can think of it as a piece of string tied in some configuration with its ends glued together. Unknot: The trivial embedding, topologically just a circle Trefoil: The simplest non-trivial knot with 3 crossings Figure-Eight: A knot with 4 crossings that is its own mirror image 2. Knot Equivalence and Ambient Isotopy An ambient isotopy is a continuous deformation of all of space that moves one knot to another without cutting or passing through itself. Think of it as smoothly manipulating a rubber band in space without breaking it. 3. Knot Invariants: Crossing and Unknotting Numbers To distinguish knots, we use invariants—quantities that remain constant under ambient isotopy. Figure-Eight: c(4₁) = 4, u(4₁) = 1 These invariants are computable but challenging—determining the unknotting number is NP-hard! This group captures the ways loops in the space around the knot can wrap around it. The knot group is a complete invariant for prime knots—it determines the knot up to mirror reflection. Wirtinger Presentation: The knot group can be computed from any diagram using generators (one per arc) and relations (one per crossing).

This is the written version of the interactive lesson above. See the full Algebraic Topology course.