BTT Mathematics / Primary Mathematics Learning Hub / Knot Theory
Knot theory studies closed loops in space and asks which deformations change only the drawing and which change the actual knot type. A mathematical knot has no loose ends. A link is a collection of two or more closed loops considered together.
A knot diagram is a flat picture with crossing information showing which strand passes over and which passes under. Different diagrams can represent the same knot because the loop may be bent or redrawn without cutting it.
This guide is Primary Mathematics enrichment. It connects Möbius Strips and Topology, Invariants, Route Networks and visual reasoning.
Formal knot polynomials are outside this guide. We use small diagrams, component count, crossing information and the three Reidemeister moves as a controlled introduction.
Knots and links · Crossing diagrams · Reidemeister moves · Simple invariants · Unknotting and crossing number · 24 questions · Worked answers
1. A knot is one closed component; a link may have several
A single unknotted circle is called the unknot. It has one component. Two disjoint circles form a two-component link, whether or not they are linked together.
Worked example A: Component count
One closed loop has 1 component. Two separate loops have 2 components.
Worked example B: Hopf link
The standard Hopf link has two components linked once around one another. It cannot be separated into two unlinked circles without cutting a component or passing one strand through another.
Worked example C: Trefoil knot
A trefoil is a one-component knot that cannot be deformed into the unknot by the allowed diagram moves alone. Its standard minimal diagram has 3 crossings.
Worked example D: Cutting changes the object
Cutting a closed loop creates loose ends and is not an allowed knot-preserving deformation.
2. A crossing count belongs to a diagram, not automatically to the knot itself
A drawing may show extra loops or twists. Some crossings can be introduced or removed without changing knot type.
Worked example E: One-curl unknot diagram
An unknot can be drawn with one apparent twist crossing. A Reidemeister-I move removes that curl, leaving a crossing-free circle. Therefore “this diagram has one crossing” does not mean the knot’s minimum crossing number is one.
Worked example F: Over/under information matters
At every diagram crossing, one strand passes over and one passes under. Erasing that information turns the knot diagram into an ambiguous plane curve.
Worked example G: Diagram crossing count
If a particular drawing contains5 crossings, its diagram crossing count is5. The underlying knot may have a smaller diagram.
Worked example H: Minimal crossing number
The crossing number of a knot is the smallest crossing count among all its diagrams. This is a knot invariant, unlike the crossing count of one arbitrary diagram.
3. Reidemeister moves change diagrams without changing knot type
Three standard local diagram moves preserve knot or link type.
Move I
Add or remove a single twist loop. The diagram crossing count changes by1.
Move II
Add or remove a pair of crossings formed by pulling one strand across another and back. The crossing count changes by2.
Move III
Slide one strand past a crossing between two others. The number of crossings stays the same, but their local arrangement changes.
Worked example I: Why these are allowed
Each move corresponds to continuously deforming the loop in three-dimensional space without cutting it and without allowing one strand to pass through another.
Worked example J: Component count under moves
None of the Reidemeister moves cuts or joins loops, so the number of components is unchanged.
Worked example K: Crossing count under Move I
A diagram with6 crossings can become one with5 after removing one removable curl. Same knot type, different diagram count.
Worked example L: Crossing count under Move II
A removable two-crossing detour changes6 crossings to4 while preserving the link type.
4. An invariant stays the same under the allowed deformations
An invariant is useful because different values prove two objects are not equivalent. The converse need not hold: equal invariant values do not always prove equivalence.
Worked example M: Number of components
A one-component knot cannot be equivalent to a two-component link because component count is invariant.
Worked example N: Minimal crossing number
The unknot has crossing number0. The trefoil has crossing number3. Therefore they are not the same knot type.
Worked example O: Diagram crossing count is not invariant
Move I changes it, so ordinary crossing count of a particular picture fails the invariant test.
Worked example P: A weak invariant
Two different knots can have the same number of components. Component count can distinguish some objects but not all.
Worked example Q: Invariant logic
If object X has1 component and object Y has2, they are different. If both have1 component, more information is needed.
5. Simplifying a diagram is a search for a simpler representative
To show a complicated diagram is an unknot, one route is to find Reidemeister moves reducing it to a crossing-free circle.
Worked example R: One removable curl
A one-crossing curl reduces by Move I to the standard unknot.
Worked example S: Two-crossing detour
If the two crossings form a Reidemeister-II pair, both disappear together.
Worked example T: A failed simplification attempt
If no immediate Move I or II is available, that does not prove the knot is nontrivial. A Move III may rearrange crossings and expose a later simplification.
Worked example U: Minimum versus current diagram
A trefoil drawn with5 crossings may still have knot crossing number3 if two crossings are removable through allowed moves.
Worked example V: Search-state viewpoint
Each diagram can be treated as a state, and each Reidemeister move as a transition. Simplification becomes a path-search problem through equivalent diagrams.
6. Practice: 24 original questions
Questions 1–8: Objects and diagrams
1. How many components does a mathematical knot have?
2. How many components does a two-loop link have?
3. What is the simplest one-component knot called?
4. Does cutting a loop preserve knot type?
5. What two pieces of information are needed at a diagram crossing?
6. A drawing has5 crossings. Is5 automatically the knot’s crossing number?
7. What is the crossing number of the unknot?
8. What is the trefoil’s minimum crossing number?
Questions 9–16: Reidemeister moves and invariants
9. Which move adds or removes one curl?
10. Which move adds or removes two crossings?
11. Which move keeps the crossing count unchanged while sliding strands?
12. Does any Reidemeister move change the number of components?
13. Is diagram crossing count a knot invariant?
14. Is number of components an invariant?
15. Can a one-component knot be equivalent to a two-component link?
16. If two links both have two components, does that prove they are equivalent?
Questions 17–24: Simplification and reasoning
17. A one-crossing curl in an unknot diagram can be removed by which move?
18. A removable two-crossing detour can be removed by which move?
19. Can a Move III expose a later simplification even though it removes no crossing itself?
20. A trefoil is drawn with5 crossings. Could its crossing number still be3?
21. What does an invariant difference prove?
22. What does equality of a weak invariant fail to prove?
23. Why is component count preserved under Reidemeister moves?
24. Describe simplification as a state-transition search.
7. Worked answers
Answers 1–8
1. One.
2. Two.
3. The unknot.
4. No.
5. Which strand passes over and which passes under.
6. No. Another diagram may use fewer crossings.
7. 0.
8. 3.
Answers 9–16
9. Reidemeister I.
10. Reidemeister II.
11. Reidemeister III.
12. No.
13. No.
14. Yes.
15. No.
16. No. Equal component count is not enough.
Answers 17–24
17. Move I.
18. Move II.
19. Yes.
20. Yes. Crossing number is the minimum over all diagrams.
21. The two objects cannot be equivalent under the allowed deformations.
22. It does not prove the objects are equivalent.
23. The moves neither cut nor join components.
24. Treat each equivalent diagram as a state and each Reidemeister move as an allowed transition; search for a path to a simpler diagram.
8. Teaching and transfer
If a learner thinks every changed drawing is a different knot, use a soft loop or cord. Bend and rotate it without cutting, then compare the dramatically different projections.
When crossing count is overused
Ask whether a Move I or II can change the count. If yes, the raw diagram count cannot be an invariant.
When an invariant is treated as a complete fingerprint
Use component count: many different knots have one component. An invariant can rule equivalence out without necessarily proving it in.
When topology feels detached from earlier work
The question is the same invariant question seen elsewhere: what may change, and what must remain fixed under the allowed transformations?
When simplification stalls
Try a rearranging move before concluding nothing can be simplified. A state may need to change form before a cancellation becomes visible.
Connect to Möbius topology
Both topics distinguish geometric appearance from deeper structure. In Möbius Strips, One-Sided Surfaces and Topology, sidedness and boundary structure survive bending; here knot type survives Reidemeister deformations.
Continue through this enrichment collection
For stochastic movement, use Random Walks, Transition Probabilities and Markov Chains. For compatible assignments, use Graph Matchings, Bipartite Assignment and Pairing Problems. For counting decompositions, use Integer Partitions, Ferrers Diagrams and Counting Structures.
Return to the BTT Primary Mathematics Learning Hub.
Original enrichment guide with 24 original practice questions and separate worked answers. Crossing count of a drawing is kept distinct from the knot invariant called crossing number.