Knots and Links

A Smooth Approach

Knot and Link

A (smooth) knot is a smooth (i.e. infinitely differentiable) embedding $K:S^1\hookrightarrow S^3$. More generally, a link $L$ with $n$ connected components is a smooth embedding of $\underset{n}{\underbrace{S^1\bigsqcup \cdots \bigsqcup S^1}}$ into $S^3$.

When are two knots same?

Homotopy is a continuous deformation, which is not enough, because one path might cross itself under the deformation. We need to consider isotopy, which is a continuous deformation of embeddings. However, it turns out that isotopy is not enough either, because any tangle can be shrunk to a point. To solve this, we need to consider one of the following:

• Ambient isotopy: a continuous deformation of the space.
• Smooth isotopy: a differentiable deformation of knots.
• Thick tubes instead of circles.

Knot Equivalence

We will say that two knots (or links) $K_1$ and $K_2$ are equivalent if they are ambient isotopic, that is, if there is a continuous $h:S^3\times [0,1]\to S^3$ such that $h(\cdot ,t)=h_t:S^3\to S^3$ satisfy $\text{DeclareMathOperator}\text{id}id{h}_0={\text{id}}_{S^3}$ and $h_1\circ K_1=K_2$. Such a map $h$ is called an ambient isotopy.

Remark

According to this definition, a knot or a link is a map. However, we often think about a knot or a link as the image of such maps, because knots with the same image are equivalent to each other.

e.g. The simplest knot is the unknot.

Knot Diagrams

Knot Diagram

A knot diagram (or link diagram) is a 4-valent graph with over/under crossing information at each vertex. The diagram is embedded in a plane $S^2\subset S^3$ called the projection plane.

Reidemeister Moves

Two knot diagrams represent the same knot if and only if one can be obtained from the other by a finite sequence of Reidemeister moves. The below are three types of Reidemeister moves:

Remark

However, if a knot diagram $D$ represents unknot, one may need a huge number of Reidemeister moves to show this. In fact, it is an open problem to determine the minimum number of Reidemeister moves required to show that a given knot diagram represents the unknot.

Algebraic Structure for Knots

Connected Sum

Connected sum is a binary operation on knots, denoted by $K_1\#K_2$, where $K_1$ and $K_2$ are two knots. The connected sum of two knots is obtained by removing a small open disk from each knot and gluing the two resulting boundaries together.

e.g.

Why connected sum is well-defined?

First, it does not matter where we attach, because, say we are connecting $K_1$ with $K_2$, we can always move $K_2$ around to grow it back. Moreover, knots are naturally equipped with a counterclockwise orientation. The connected sum of two knots is well defined up to orientation. i.e. there is only one way to connect in an orientation-preserving way, and we can only pick one of the following:

Proposition

Connected sum is associative, unital and commutative. Thus the set of all knots with the connected sum operation forms a commutative monoid, denoted by $(\mathcal{K},\#)$.

Proof Clearly the unit is the unknot, commutativity and associativity are inherited from the underlying operation.

Since such monoid is just isomorphic to positive integers with multiplication $({\mathbb{Z}}_{>0},\cdot )$, we can introduce the following concepts accordingly:

Prime Knot

A knot is prime if it is not the connected sum of two non-trivial knots.

Theorem

Every knot has a unique prime decomposition. There are infinitely many prime knots.

Proof By the isomorphism. $\square$