Knot Theory — When Is a Knot Really the Same Knot?
Anyone can tie a knot. Proving which knot you have tied is considerably harder.
Give a student a piece of string and ask them to tie a knot.
It sounds like something for a five-year-old rather than a mathematics lesson.
Now give them two pieces of string, tie a different-looking knot in each, and ask:
Are these actually different knots?
That question is considerably harder.
You can pull them. Twist them. Stretch them. Turn them over. Move one loop through another.
But there is one important rule:
You are not allowed to cut the string.
Suddenly, playing with string has become mathematics.
Welcome to knot theory, a branch of topology in which mathematicians try to understand knots, links and tangled structures.
And one of its most fundamental questions is wonderfully simple to state:
How can we prove that two knots are different?
First, There Is a Problem With Ordinary String
If I take a normal piece of string, tie a knot in it and leave the two ends free, I can usually undo the knot by manipulating one of the ends.
Mathematicians therefore do something slightly different.
Imagine joining the two ends of the string together.
We now have a closed loop.
If the loop contains no knot, we have the simplest possible mathematical knot:
the unknot.
It is essentially just a loop.
Now suppose we tie a trefoil knot in the string before joining the ends.
We have produced something fundamentally different.
The question is:
How do we know?
It might look different, but appearances can be deceptive.
Topology — Mathematics Without a Ruler
Knot theory belongs to a larger area of mathematics called topology.
Topology is sometimes described informally as the mathematics of shape when distances and angles do not matter.
Imagine an object made from perfectly flexible rubber.
You are allowed to:
stretch it;
squash it;
bend it;
twist it.
But you cannot:
tear it;
cut it;
glue previously separate pieces together;
pass one part magically through another.
Under those rules, many objects that look different geometrically are actually regarded as the same topological object.
This is a very different way of thinking from the geometry students normally meet at school.
In geometry, changing a square into a rectangle matters because the lengths and angles have changed.
In topology, many such changes are irrelevant.
What matters is the underlying structure.
And that is exactly the idea we need for knots.
Experiment 1 — Can You Fool Someone With the Unknot?
Start with a simple closed loop of cord.
Now twist and distort it.
Lay it on a table so that one part crosses another several times.
Ask someone:
Is this knotted?
It may look extremely complicated.
But if you can continuously manipulate it back into a simple loop without cutting it or allowing the string to pass through itself, then mathematically it was still the unknot.
This immediately teaches an important mathematical lesson:
Complex appearance does not necessarily mean complex structure.
That principle occurs throughout mathematics.
Knot Diagrams — Turning String Into Mathematics
Working with actual string is useful, but mathematicians need a way of recording knots on paper.
They use a knot diagram.
Imagine placing the knot on a table and looking directly down at it.
Where one part of the string crosses another, we indicate which strand passes over and which passes underneath.
This distinction is crucial.
Two diagrams may contain exactly the same pattern of crossings but represent different knots if the over-and-under information changes.
This gives students an immediate practical activity.
Tie a knot.
Place it on white paper.
Photograph it from above.
Then draw its knot diagram.
Suddenly a three-dimensional object has become a two-dimensional mathematical representation.
The First Three Knots to Investigate
A very good progression is:
unknot → trefoil knot → figure-eight knot
1. The Unknot
This is the simplest possible closed loop.
Its minimum number of crossings is:
0
Of course, you can draw the unknot with lots of apparent crossings by twisting it around.
But those crossings can ultimately be removed.
That word minimum is important.
2. The Trefoil Knot
The trefoil is probably the most famous non-trivial knot.
Its simplest diagram has:
3 crossings
Try as you might, you cannot reduce it to a diagram containing zero crossings without cutting the loop or allowing one strand to pass through another.
That makes it fundamentally different from the unknot.
3. The Figure-Eight Knot
The figure-eight knot has a minimum of:
4 crossings
It is another genuinely different knot.
Now we have the beginnings of a classification system.
But we immediately encounter a mathematical difficulty.
Is counting crossings enough to identify a knot?
Unfortunately, no.
Why Counting Crossings Isn't Enough
Suppose I take a trefoil and twist part of it so that my diagram contains five crossings.
Has the trefoil suddenly become a five-crossing knot?
No.
The underlying knot has not changed.
We therefore distinguish between:
the number of crossings in a particular diagram
and
the minimum crossing number of the knot itself.
The latter is an example of a property that can help classify knots.
This leads to one of the central ideas of knot theory:
invariants.
What Is an Invariant?
An invariant is something that remains unchanged when we manipulate the knot in permitted ways.
Students already know invariants, even if they have never used the word.
For example, suppose I rearrange:
2 + 7 + 3
as:
7 + 3 + 2
The order has changed, but the total remains 12.
The total is invariant under that rearrangement.
In knot theory, mathematicians search for properties that remain unchanged when the knot is stretched and manipulated.
If two knots have different values for a suitable invariant, they cannot be equivalent.
This is a wonderfully important mathematical strategy:
Don't try to compare every detail. Find something that cannot change.
But How Do We Know When Manipulating a Diagram Is Legal?
This is where three wonderfully elegant operations enter the story.
They are called the Reidemeister moves.
They describe changes we can make to a knot diagram without changing the underlying knot.
Reidemeister Move I — The Twist
Imagine making a little loop in part of the string.
On the diagram, this introduces a crossing.
Undo the loop and the crossing disappears.
The knot itself has not changed.
This immediately explains why simply counting the visible crossings in a diagram cannot identify a knot.
You can create extra crossings just by twisting the string.
Reidemeister Move II — Two Crossings Appear or Disappear
Imagine pulling one section of string across another so that two crossings appear.
Reverse the movement and both disappear.
Again, the knot itself has not changed.
The diagram looks different.
The topology does not.
Reidemeister Move III — Sliding a Strand
The third move involves sliding one strand past a crossing formed by two other strands.
No crossings are created or destroyed.
Their arrangement simply changes.
This one can be harder to visualise on paper, so actual cord or rope makes a much better teaching tool.
And then comes a remarkable result:
Two diagrams represent the same knot if one can be transformed into the other using Reidemeister moves.
That is an extraordinarily powerful statement.
A Practical Challenge — Same Knot or Different Knot?
This can become a superb classroom or home-laboratory investigation.
Prepare several loops of differently coloured cord.
Some should be:
unknots disguised with twists;
trefoils arranged in different ways;
figure-eight knots;
mirror-image trefoils.
Label them A, B, C, D and so on.
Then ask students to classify them.
But there is an important rule.
They cannot simply say:
"They look the same."
They must provide mathematical evidence.
Can one be transformed into another?
Can crossings be removed using Reidemeister moves?
Is there an invariant that distinguishes them?
This changes the activity from puzzle solving into mathematical reasoning.
A Particularly Interesting Question — What About a Mirror?
Tie a trefoil.
Now imagine looking at it in a mirror.
Is the reflected knot the same knot?
This introduces another beautiful idea.
Some knots have chirality, or handedness.
A trefoil can be right-handed or left-handed.
The two are mirror images.
But one cannot simply be manipulated into the other using the permitted transformations.
It is rather like your hands.
Your left hand and right hand have the same basic structure.
One is the mirror image of the other.
But no matter how you rotate your left hand in ordinary three-dimensional space, you cannot turn it into a right hand.
That makes the trefoil an excellent bridge between physical intuition and abstract mathematics.
Could We Build a Knot Detective?
This is where the subject becomes even more interesting for students studying computing.
Imagine taking photographs of different knots.
Could a computer determine which knot it was looking at?
At first it sounds straightforward.
Count the crossings.
But we have already discovered the problem.
A trefoil can be drawn with more than three crossings.
An unknot can be made to look extremely complicated.
So the computer would need something more sophisticated.
It would need to recognise transformations.
It might need to calculate invariants.
It might need algorithms capable of deciding whether two knot diagrams represent the same underlying knot.
A simple piece of string has suddenly connected:
mathematics → topology → algorithms → computer science.
From GCSE Mathematics to University Mathematics
One reason I particularly like topics such as knot theory is that they challenge the idea that mathematics is simply a ladder of increasingly difficult calculations.
At GCSE, mathematics can sometimes appear to consist of:
solving equations;
calculating percentages;
manipulating fractions;
finding angles;
drawing graphs.
All of those are important.
But they are not the whole of mathematics.
Knot theory requires surprisingly little conventional calculation to get started.
A GCSE student can understand the central problem:
Are these two knots really the same?
Yet answering that question completely can require extremely sophisticated mathematics.
That makes it ideal for showing students what mathematics looks like beyond the examination syllabus.
Mathematics Begins With Definitions
There is another lesson hidden inside knot theory.
Mathematicians have to be extraordinarily careful about definitions.
What exactly counts as the same knot?
What transformations are allowed?
What does "crossing" mean?
What properties remain invariant?
This is not mathematical pedantry.
Without precise definitions, we cannot prove anything.
Consider two students looking at two pieces of string.
One says:
"They are obviously the same knot."
The other says:
"No they aren't."
Who is correct?
Looking is not enough.
We need a definition of equivalence and a method for establishing it.
That is mathematics.
What Does "Proof" Mean Here?
This leads to perhaps the most valuable discussion of all.
Students often encounter proof through algebra or geometry.
Prove that two angles are equal.
Prove that an expression is divisible by something.
Prove an identity.
Knot theory offers a different style of proof.
Suppose I claim that two knot diagrams represent the same knot.
One possible proof is constructive:
I show a sequence of permitted transformations that converts one into the other.
But suppose I claim they are different.
That is harder.
I cannot simply say:
"I tried for ten minutes and couldn't untangle it."
Failure to find a transformation is not proof that no transformation exists.
Instead, we need a property that distinguishes them.
That is where invariants become so powerful.
If invariant K has one value for knot A and a different value for knot B, then the knots cannot be equivalent.
This is a profound mathematical strategy:
Find something that must stay the same. Then show that it doesn't.
Going Further — Knot Polynomials
For GCSE students, I would probably stop before the algebra becomes too sophisticated.
For A-level and Further Maths students, however, we can mention that mathematicians have developed algebraic objects associated with knots.
Examples include:
the Alexander polynomial;
the Jones polynomial;
the HOMFLY polynomial.
The details quickly lead beyond school mathematics.
But the principle is accessible.
We take a geometric object — a knot — and associate an algebraic object with it.
Then we use algebra to investigate topology.
This is one of the recurring themes of higher mathematics:
Translate a difficult problem into another mathematical language where it becomes easier to study.
And Then There Is DNA
There is a fascinating real-world connection.
DNA molecules are extremely long compared with their width and can become twisted, looped and tangled.
Certain enzymes cut, pass and reconnect strands of DNA.
That means questions about knotting and linking become relevant when studying the geometry and topology of DNA.
Knot theory has therefore found applications in areas of molecular biology.
I would keep this secondary when teaching the topic because the mathematics is interesting enough in its own right.
But it provides a wonderful answer when a student inevitably asks:
"Does anyone actually use this?"
Yes.
What began as playing with loops of string can help provide mathematical tools for describing tangled molecules.
Try This at Home
You need very little equipment:
several pieces of cord or thick string;
tape or a method of joining the ends;
paper;
a pen;
ideally a camera or phone.
Start with the unknot.
Then make a trefoil.
Then make a figure-eight knot.
Photograph each from above and draw its knot diagram.
Next, deliberately make each knot look more complicated without changing the underlying knot.
Add twists.
Change its shape.
Turn it over.
Then challenge someone else to identify it.
Finally, try producing two very complicated-looking diagrams that are actually both unknots.
The lesson becomes:
Never trust the diagram alone.
A Further Maths Challenge
Here is a question worth thinking about.
Suppose two knots have the same minimum crossing number.
Does that prove they are the same knot?
No.
So crossing number is useful, but not sufficient.
This raises the next question:
What collection of invariants would be sufficient to identify every knot uniquely?
Now we are no longer asking an examination-style question.
We are beginning to think like mathematicians.
Why I Would Teach This Even Though It Isn't on the Syllabus
I think students should occasionally encounter mathematics for which there is no examination mark attached.
Knot theory is ideal.
The equipment costs almost nothing.
There is no long list of formulae to learn.
A student can begin investigating it almost immediately.
Yet underneath that simple activity lie some of the most important ideas in mathematics:
abstraction;
classification;
equivalence;
invariants;
transformation;
proof;
algorithms;
topology.
Most importantly, it challenges the idea that mathematics is primarily about calculating answers.
Sometimes mathematics asks something much deeper:
How do we know that two things are fundamentally the same?
And equally importantly:
How could we prove that they are not?
From a Piece of String to Modern Mathematics
That is what makes knot theory such a wonderful piece of mathematics beyond the syllabus.
Begin with a piece of string.
Tie a knot.
Join the ends.
Draw it.
Twist it.
Stretch it.
Try to simplify it.
Then ask one apparently innocent question:
What knot is it?
The unknot seems obvious.
The trefoil looks manageable.
The figure-eight introduces another possibility.
Then come mirror images, invariants, Reidemeister moves and knot polynomials.
Before long, the student who thought they were simply playing with string has encountered genuine topology.
And that is exactly why I enjoy taking students beyond the examination specification.
The syllabus teaches us mathematical techniques.
But occasionally stepping outside it shows us something equally important:
what mathematics actually is.
Anyone can tie a knot.
Proving which knot you have tied is considerably harder.
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