Infinite Knots

Alex Westphal · 23 Sep 2014

Sometimes I hear people claim “I can tie every knot”. In general I can disabuse them of this notion by demonstrating a knot that they don’t know (I know 60+, most of which are listed on the knot reference page).

If the claim was from someone that knew more knots than I do (uncommon but easily possible), we can turn to The Ashley Book of Knots. It is highly improbable that even Clifford W Ashley could tie all 3800+ knots listed therein without referring to some form of documentation.

Going even further, there are theoretically infinitely many knots. In this article we will look at a proof of this, realised through twisting an overhand knot.

The Natural Numbers

The set of th natural numbers $\mathbb{N} = \{0,1,2,3,4,\ldots\}$ (which will serve as a convenient proxy for our knots) form an infinite set. Mathematically this can be denoted as:

$$\forall \ n \in \mathbb{N} \ \exists \ n+1 \in \mathbb{N}$$

That is, for all $n$ in the set $\mathbb{N}$ there exists $n+1$ that is also a member of the set $\mathbb{N}$.

Numbering Knots

We will describe the set of knots formed by twisting an overhand knot as $\mathbb{K} = \{k_0,k_1,k_2,k_3,k_4,\ldots\}$ where $k_0$ is an un-knotted piece of rope, $k_1$ is the simple overhand knot, and $k_x$ is an overhand knot with $x$ turns.

Interestingly the set of knots $\mathbb{K}$ is isomorphic to the set of natural number $\mathbb{N}$. That is, there is a two way mapping between $\mathbb{K}$ and $\mathbb{N}$ ($n \in \mathbb{N} \leftrightarrow k_n \in \mathbb{K}$).

Visually

A table of the the knots of the first 7 knots:

$k_x$	Common Name	Diagram
$k_0$	Bight
$k_1$	Overhand Knot
$k_2$	Figure Eight Knot
$k_3$	Stevedore Knot
$k_4$
$k_5$
$k_6$

I hope you can see that for any knot $k_n$, adding extra twists yields $k_{n+1}, k_{n+2}, \ldots$ and thus by the isomorphism with the natural numbers, there are infinite number of knots.