Knot groups and the connection to group theory
The fundamental group π₁(S³ - K) is the ultimate knot invariant - it contains complete information about the knot's topology. Unlike polynomial invariants which can fail to distinguish different knots, the fundamental group is a complete invariant: if two knots have isomorphic fundamental groups, they are equivalent!
In this module, you'll learn how to visualize loops in the knot complement, compute knot groups from diagrams, work with group presentations, and discover the deep connections to the Group Theory module. This is where topology and algebra unite in beautiful harmony.
The fundamental group π₁(S³ - K) consists of equivalence classes of loops in 3D space that avoid the knot K. Two loops are equivalent if one can be continuously deformed into the other without crossing the knot.
Description: A loop that goes around the knot once
Contractible: ✗ No
Reason: Cannot be shrunk to a point - the knot is in the way!
| Loop Type | Contractible? | Group Element |
|---|---|---|
Simple Loop | ✗ | a |
Trivial Loop | ✓ | e (identity) |
Double Loop | ✗ | a² |
Linked Loop | ✗ | b |
The group operation is concatenation of loops. To multiply two loops, traverse the first loop, then the second, both starting and ending at the base point.
The fundamental group captures the topology of the knot complement - the space around the knot. It consists of equivalence classes of loops based at a point.
Key properties:
For each knot K, we can compute its knot group π₁(S³ - K), the fundamental group of the knot complement. This is given as a group presentation with generators and relations.
2 generators
1 relation(s)
The abelianization H₁(S³ - K) is the largest abelian quotient of the knot group. It's obtained by making all generators commute.
Remarkably, the abelianization of any knot group is ℤ! This is because every knot has a Seifert surface, and the linking number with that surface generates the first homology.
| Knot | Presentation | Abelian? |
|---|---|---|
| Unknot (0₁) | ⟨a | ⟩ ≅ ℤ | ✓ |
| Trefoil (3₁) | ⟨a, b | aba = bab⟩ | ✗ |
| Figure-Eight (4₁) | ⟨a, b | aba⁻¹b⁻¹aba = bab⁻¹a⁻¹bab⟩ | ✗ |
| Cinquefoil (5₁) | ⟨a, b | a²ba²b = ba²ba²⟩ | ✗ |
| Three-Twist (5₂) | ⟨a, b | aba⁻¹b = ba⁻¹ba⁻¹⟩ | ✗ |
The Wirtinger presentation is an algorithmic method to compute the knot group from a knot diagram:
xax⁻¹ = bThe fundamental group is a complete invariant for knots: if two knots have isomorphic fundamental groups, they are equivalent!
Key facts:
Learn how to work with group presentations using Tietze transformations. These operations preserve the isomorphism class while simplifying the presentation.
Directly from the knot diagram - often has redundant generators and relations
Use first relation to substitute x₃ = x₁x₂x₁⁻¹
Multiply both sides by x₁ on the right
Standard form: a = x₁, b = x₂
This is the Braid group B₃
The allowed operations that preserve the group isomorphism class:
| Example | Original Generators | Final Generators | Result |
|---|---|---|---|
| Trefoil Simplification | 3 | 2 | Braid group B₃ |
| Unknot Simplification | 2 | 1 | Infinite cyclic group |
| Hopf Link | 2 | 2 | Free abelian group (rank 2) |
Group presentations are a powerful way to describe groups, but working with them can be tricky:
The fundamental group connects knot theory to group theory. Every concept you learned in the Group Theory module appears here! This demonstrates the deep unity of mathematics.
Knot groups are given as presentations ⟨generators | relations⟩ from Wirtinger method
General way to describe groups abstractly using generators and defining relations
Trefoil: ⟨a, b | aba = bab⟩ is a presentation of the braid group B₃
| Group Theory Concept | Knot Theory Realization | Module |
|---|---|---|
| Group Presentations | Knot groups are given as presentations ⟨generators | relations⟩ from Wirtinger method... | Link → |
| Abelianization | Every knot group has abelianization H₁(S³ - K) ≅ ℤ... | - |
| Group Homomorphisms | Knot invariants like the Alexander polynomial come from homomorphisms π₁(S³-K) → ℤ[t... | Link → |
| Free Groups | Wirtinger presentation starts with a free group on arc generators before imposing crossing relations... | - |
| Quotient Groups | Knot group G = F/N where F is free group on arcs and N is normal subgroup generated by relations... | - |
| Group Isomorphism | Two knots are equivalent if and only if their fundamental groups are isomorphic... | Link → |
| Braid Groups | Torus knots have fundamental groups that are braid groups or quotients thereof... | Link → |
| Commutators | Commutator subgroup [G... | - |
The fundamental group demonstrates how different areas of mathematics are deeply connected:
The fundamental group is a perfect example of how abstract algebra (group theory) and geometric topology (knot theory) inform each other:
Two knots K₁ and K₂ are equivalent if and only if π₁(S³ - K₁) ≅ π₁(S³ - K₂). The fundamental group is a complete invariant for knots.
For any classical knot K, the abelianization H₁(S³ - K) ≅ ℤ. This follows from the existence of a Seifert surface and the linking number invariant.
Given a knot diagram with n arcs, the fundamental group has a presentation with n generators (one per arc) and n relations (one per crossing). At each crossing where arc x goes over arcs a and b, the relation is xax⁻¹ = b.
A knot is determined by its complement up to reflection. That is, if S³ - K₁ ≅ S³ - K₂, then K₁ and K₂ are equivalent or mirror images. This profound result shows the complement contains all the information!
Given a knot diagram, the Wirtinger presentation can be computed in linear time. It's a purely mechanical process.
Given a presentation ⟨S | R⟩ and two words w₁, w₂, determining if they represent the same group element is undecidable in general! For knot groups specifically, practical algorithms exist but no general solution.
Given two presentations, determining if they describe isomorphic groups is undecidable. This means we can't algorithmically verify knot equivalence using fundamental groups alone!
In practice, we extract computable invariants (like polynomial invariants!) from the fundamental group. These provide practical tests even though the theoretical problems are undecidable.
1. Fundamental Group: We visualized loops in the knot complement and saw how they form a group under concatenation. The unknot has π₁ = ℤ, while the trefoil has the more complex braid group ⟨a, b | aba = bab⟩.
2. Wirtinger Presentations: We learned the algorithmic Wirtinger method to compute knot groups from diagrams: one generator per arc, one relation per crossing. Every knot group has a finite presentation!
3. Group Presentations: We practiced simplifying presentations using Tietze transformations, reducing the trefoil's 3-generator Wirtinger presentation to the elegant 2-generator form ⟨a, b | aba = bab⟩.
4. Connection to Group Theory: We discovered how every concept from group theory (presentations, homomorphisms, quotients, abelianization, free groups) appears naturally in knot theory. This is mathematics at its most unified!
Next Up: You've now explored the core theory! Ready for advanced topics? In the next phase, you'll discover Seifert surfaces, knot genus, fibered knots, and hyperbolic geometry - the cutting edge of modern knot theory.