Knots, Links, and Diagrams - Core Definitions

Knot theory studies the mathematical properties of knots and links embedded in three-dimensional space. The foundational concepts provide the language for all subsequent theory.

Definition

A knot is a smooth embedding $K: S^1 \to \mathbb{R}^3$ (or $S^3$ ) of a circle into three-dimensional space. Two knots $K_1$ and $K_2$ are equivalent or ambient isotopic if there exists a continuous family of homeomorphisms $h_t: \mathbb{R}^3 \to \mathbb{R}^3$ for $t \in [0,1]$ with $h_0 = \text{id}$ and $h_1(K_1) = K_2$ .

The unknot or trivial knot is any knot equivalent to a standard planar circle.

The notion of ambient isotopy captures the intuitive idea that we can deform one knot into another through continuous motion without cutting or passing strands through each other. This equivalence relation partitions all knots into equivalence classes, which are the primary objects of study in knot theory.

Definition

A link of $n$ components is a smooth embedding $L: S^1 \sqcup \cdots \sqcup S^1 \to \mathbb{R}^3$ of $n$ disjoint circles. A link with one component is simply a knot. The linking number $\text{lk}(L_1, L_2)$ measures how two components wind around each other.

Definition

A knot diagram is a regular projection $\pi: K \to \mathbb{R}^2$ of a knot onto a plane such that:

Only finitely many double points occur (crossings)
No triple or higher points occur
At each crossing, over/under information is specified

The crossing number $c(K)$ is the minimum number of crossings in any diagram of $K$ .

Knot diagrams are the practical representation we use to study knots on paper. The crucial data at each crossing determines which strand passes over the other, typically indicated by breaking the under-crossing strand.

Example

Classical examples include:

The trefoil knot $3_1$ : simplest nontrivial knot with $c(3_1) = 3$
The figure-eight knot $4_1$ : crossing number $c(4_1) = 4$
The Hopf link $2_1^2$ : two linked circles with linking number $\pm 1$
The Borromean rings: three mutually unlinked circles that cannot be separated

The trefoil cannot be deformed to the unknot, proving that nontrivial knots exist. Its chirality (handedness) distinguishes left and right trefoils.

Remark

Two fundamental questions drive knot theory:

Recognition problem: Given two diagrams, are they equivalent?
Classification problem: Enumerate all distinct knot types

Both problems remain computationally difficult. Knot tables classify knots by increasing crossing number, with modern tables extending beyond 16 crossings containing millions of distinct knots.

The mathematical formalization distinguishes between the abstract knot (an equivalence class) and its diagrams (concrete representations). While a single knot has infinitely many diagrams, Reidemeister moves provide a systematic way to transform between them, forming the basis for defining knot invariants that remain unchanged under these transformations.