Geometric Topology 2

Knots

Recall $S^1 = \{(\cos(t),\sin(t)): t \in [0,2\pi)\} \subset \mathbb{R}^2$.

Definition:

A knot is the image of a continuous injective mapping $f: S^1 \rightarrow \mathbb{R}^3$. We will assume that $K = f(S^1)$ can be surrounded by a tube of some fixed radus without intersections (this means $K$ is "tame").

So $f: S^1 \rightarrow K$ is bijective, it follows since $S^1$ is compact and $\mathbb{R}^3$ is Hausdorff that $f^{-1}$ is continuous, so $S^1$ and $K$ are homeomorphic.

A table of the prime knots with $7$ or fewer crossings

Corollary: Any two knots are homeomorphic (to a circle).

Let $K = f(S^1)$ be a (tame) knot in $\mathbb{R}^3$. We can project $K$ orthogonally onto any plane. Such a projection gives a valid diagram if it is 1:1 apart from over a finite number $c \geq 0$ of points of $\mathbb{R}^2$ where it is 2:1 and the branches are transversal (i.e. no "cusps" or triple-meeting points, just one crossing point for each strand of the knot). These are called crossings, each of which has an underpass and an overpass.

A "wild" knot, which we will ignore and instead focus on "tame" knots.

Definitions:

An arc is a strand from an underpass to an underpass. If a diagram has $c$ crossings it has $c$ arcs. A bridge is an arc with at least one overpass.

Definition:

The writhe-sign of a crossing in a knot diagram is defined as the the following:

• We assign an arbitrary "orientation" to the knot diagram by writing arrows in a certain direction along the path of the diagram until one returns to the point from which one started.
• A sign of $+1$ is given when a crossing has the "positive $x$"-axis of the crossing as an overpass and the "positive $y$"-axis of the crossing therefore as an underpass. We can show this by rotating the crossing around until it is oriented with one strand in the positive $x$-direction and one in the positive $y$-direction.
• A sign of $-1$ is given when the crossing has the positive $x$-axis as an underpass and the positive $y$-axis as an overpass.

The writhe sign of a crossing.

Definition:

The writhe of a knot diagram $D$ with $c$ crossings is therefore defined as:

Definition:

A link $L$ is a disjoint union of $\ell$ knots in space. We say that $L$ has $\ell$ components.

Some knots and links.

Remarks:

• Any knot is a link with one component
• If $K_i, K_j$ are two oriented knots with diagrams $D_i,D_j$ then their linking number is:
  $$\ell_{i,j} = \frac{1}{2}\sum_x \text{writhe-sign}(K_i,K_j)$$
  where $x$ runs over all crossings between $D_i$ and $D_j$.

Definition:

Ambient isotopy is the "natural" notion of equivalence of knots in space. There are equivalent definitions:

1. $K_0,K_1$ are ambient isotopic if $K_0$ can be manouvered by hand in space to a possibly rescaled $K_1$.
2. $K_0,K_1$ are ambient isotopic if there exists a homomorphism $F: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ that preserves orientation taking $K_0$ to $K_1$, i.e. $F(K_0) = K_1$.

Let $K_0,K_1$ be knots in space. Choose $D_0,D_1$ as diagrams to represent $K_0,K_1$.

Question: How do we know from $D_0,D_1$ whether $K_0,K_1$ are equivalent?

Definition: Two diagrams in the plane are called isotopic if they represent knots that are ambient isotopic in space.