REFERENCE SHEET

Knots, links & invariants

A knot is an embedding, not just a drawing with crossings.

knot group = π₁(S³ ∖ K)

Definitions

Knot and link

A knot is a smooth (or tame) embedding of S¹ in S³; a link is an embedding of finitely many disjoint circles. Equivalence here is ambient isotopy.

Diagram

A generic planar projection with over/under information at crossings.

Invariant

A quantity or algebraic object unchanged under the chosen equivalence. Different invariant values distinguish knots; equal values need not identify them.

Key theorems

Reidemeister theorem

Two diagrams represent ambient-isotopic tame links exactly when related by planar isotopy and finitely many Reidemeister moves of types I, II, and III.

Knot-group invariance

Ambient-isotopic knots have homeomorphic complements and therefore isomorphic fundamental groups.

Unknot complement

The fundamental group of the unknot complement in S³ is ℤ; the complement of an open tubular neighborhood is a solid torus.

Notable applications

Embedded geometry

Distinguish embeddings of the same abstract circle that cannot be deformed into one another without crossings.

Molecular topology

Closed polymer and DNA models use knot and link types to describe entanglement.

Work it out

Try before revealing

01 / Explain

Does adding one removable curl to an unknot diagram change its knot type?

Hint

Identify the Reidemeister move.

Solution

No. A type-I move adds or removes such a curl by ambient isotopy. It changes diagram crossing count without changing the knot.

02 / Reason

If a knot has a nonabelian knot group, can it be the unknot?

Hint

Compare with the unknot group.