Knot groups and the connection to group theory
The fundamental group is the ultimate knot invariant. 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. It captures all topological information about the knot complement.
This module covers how to visualize loops in the knot complement, compute knot groups from diagrams using the Wirtinger method, work with group presentations and Tietze transformations, and discover the deep connections between knot theory and group theory.
The fundamental group consists of equivalence classes of loops in 3D space that avoid the knot. Two loops are equivalent if one can be continuously deformed into the other without crossing the knot. The group operation is concatenation of loops.
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.
Key insight: The unknot has fundamental group isomorphic to the integers (every loop wraps around some number of times), while the trefoil has a non-abelian group. The non-commutativity of the trefoil group reflects the genuine complexity of its knotting.
For each knot K, we can compute its knot group as a group presentation with generators and relations. The Wirtinger method gives one generator per arc and one relation per crossing in the knot diagram.
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 = bKey insight: The fundamental group is a complete invariant -- two knots are equivalent if and only if their groups are isomorphic. However, determining whether two group presentations give isomorphic groups is undecidable in general, so in practice we extract computable invariants from the group.
Learn how to simplify group presentations using Tietze transformations. These operations preserve the isomorphism class while reducing the number of generators and relations, transforming a raw Wirtinger presentation into a clean, minimal form.
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) |
Key insight: The trefoil's 3-generator Wirtinger presentation simplifies to the elegant 2-generator form 〈a, b | aba = bab〉, which is the braid group B₃. Tietze transformations are the algebraic analogue of Reidemeister moves in topology.
The fundamental group connects knot theory to group theory. Every concept from group theory -- presentations, homomorphisms, quotients, abelianization, free groups -- appears naturally in the study of knots. 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:
Key insight: Every knot gives you an infinite non-abelian group (except the unknot, whose group is cyclic). Group presentations have concrete geometric meaning via knot diagrams, and techniques from each field cross-pollinate: Tietze transformations in algebra correspond to Reidemeister moves in topology.