Knots, Links, and Diagrams - Key Properties

The structural properties of knots and their diagrams reveal deep connections between topology, algebra, and combinatorics.

Definition

A knot $K$ is alternating if it has a diagram where crossings alternate between over and under as one traverses the knot. A knot is prime if it cannot be decomposed as a connected sum of two nontrivial knots. The bridge number $b(K)$ is the minimum number of local maxima in any height function on $K$ .

Alternating knots form a distinguished class with special properties. The alternating trefoil and figure-eight knots contrast with non-alternating knots like $8_{19}$ , which has no alternating diagram despite minimal crossing number 8.

Definition

For a knot diagram $D$ , define:

Writhe $w(D) = \sum_{\text{crossings}} \epsilon_i$ where $\epsilon_i = \pm 1$ depends on crossing orientation
Tricolorability: a diagram is tricolorable if strands can be colored with three colors such that at each crossing, either all three strands have the same color or all three have different colors

Example

The trefoil knot is tricolorable, providing a simple proof it's nontrivial. Assign colors red, blue, green to the three strands of a minimal diagram. At each crossing, the three incoming strands have different colors, satisfying the tricolorability condition. Since the unknot is not tricolorable (any coloring uses only one or two colors), the trefoil cannot be equivalent to the unknot.

Remark

The writhe is not a knot invariant (it changes under Reidemeister move I), but it plays a crucial role in defining the Jones polynomial. Tricolorability generalizes to $p$ -colorability for any prime $p$ , connecting knot theory to modular arithmetic and group theory.

Definition

The mirror image $K^*$ of a knot $K$ is obtained by reversing all crossing orientations in any diagram. A knot is achiral or amphichiral if $K \cong K^*$ . Otherwise, $K$ is chiral.

The reverse $-K$ is obtained by reversing the orientation of the knot. A knot is reversible if $K \cong -K$ .

Chirality captures an important geometric property: the left-handed trefoil $3_1$ and right-handed trefoil $3_1^*$ are distinct knots, provable using the Jones polynomial. The figure-eight knot $4_1$ is achiral, a rare property among knots.

Example

Consider the knot complement $X_K = S^3 \setminus K$ . Key properties include:

$X_K$ is a connected, orientable 3-manifold
$\pi_1(X_K)$ , the knot group, is a complete invariant for prime knots
The boundary $\partial X_K$ is a torus $S^1 \times S^1$

For the unknot, $X_{\text{unknot}} \cong S^1 \times D^2$ (solid torus), so $\pi_1(X_{\text{unknot}}) \cong \mathbb{Z}$ .

The knot complement transforms knot theory into 3-manifold topology. Thurston's geometrization program showed that most knot complements admit hyperbolic structures, connecting knot theory to differential geometry and leading to powerful computational tools for distinguishing knots.