Jones, Alexander, and HOMFLY-PT polynomials - the computational crown jewels
In 1984, Vaughan Jones discovered a revolutionary polynomial invariant that transformed knot theory and earned him the Fields Medal. The Jones polynomial can distinguish knots that previous invariants could not, opening new connections to physics, algebra, and quantum computing.
In this module, you'll explore the Jones polynomial and the classical Alexander polynomial, compare their strengths and weaknesses, and see how polynomial invariants succeed where simpler invariants fail. These tools are the most powerful instruments we have for distinguishing and classifying knots.
The Jones polynomial V(t) is a revolutionary knot invariant discovered by Vaughan Jones in 1984. It's a Laurent polynomial that can distinguish many knots that other invariants cannot!
| Knot | Jones Polynomial V(t) | V(1) |
|---|---|---|
| Unknot (0₁) | 1 | 1 |
| Trefoil (3₁) | t + t³ - t⁴ | 1 |
| Trefoil (Mirror) (3₁*) | t⁻¹ + t⁻³ - t⁻⁴ | 1 |
| Figure-Eight (4₁) | -t² + 1 - t⁻² + t⁻³ - t⁻⁴ | 0 |
| Cinquefoil (5₁) | t² + t⁴ + t⁶ - t⁷ - t⁸ | 1 |
| Three-Twist (5₂) | t + t³ + t⁵ - t⁶ | 2 |
If a knot K has Jones polynomial VK(t), then its mirror image K* has polynomial VK*(t) = VK(t⁻¹).
Vaughan Jones discovered this polynomial in 1984 while studying von Neumann algebras. It revolutionized knot theory and earned him the Fields Medal in 1990!
Key properties:
The Alexander polynomial Δ(t) is the classical polynomial invariant, discovered by J.W. Alexander in 1928. While less powerful than the Jones polynomial, it's easier to compute and has beautiful algebraic properties.
The determinant of a knot is |Δ(-1)|, the absolute value of the Alexander polynomial evaluated at t = -1.
One of the first non-trivial knots analyzed by Alexander in 1928.
| Knot | Alexander Polynomial Δ(t) | Det |
|---|---|---|
| Unknot (0₁) | 1 | 1 |
| Trefoil (3₁) | t - 1 + t⁻¹ | 3 |
| Figure-Eight (4₁) | -1 + 3t - t² | 5 |
| Cinquefoil (5₁) | t² - t + 1 - t⁻¹ + t⁻² | 5 |
| Three-Twist (5₂) | 2t - 3 + 2t⁻¹ | 7 |
| Stevedore (6₁) | 2t - 3 + 2t⁻¹ | 7 |
The Alexander polynomial cannot distinguish all knots. For example, 5₂ and 6₁ have the same polynomial:
This is where the Jones polynomial excels - it CAN distinguish these knots!
The Alexander polynomial was the first polynomial invariant discovered. It comes from the homology of the infinite cyclic cover of the knot complement.
Key properties:
Different polynomial invariants have different strengths. Compare the Jones, Alexander, and HOMFLY polynomials side-by-side to see which ones can distinguish different knots.
First non-trivial knot - distinguished by all polynomials.
| Knot | Jones V(t) | Alexander Δ(t) | HOMFLY P(a,z) |
|---|---|---|---|
| 0₁ | 1 | 1 | 1 |
| 3₁ | t + t³ - t⁴ | t - 1 + t⁻¹ | a² + a²z² |
| 4₁ | -t² + 1 - t⁻² + t⁻³ - t⁻⁴ | -1 + 3t - t² | a² - 2 + a⁻² |
| 5₂ | t + t³ + t⁵ - t⁶ | 2t - 3 + 2t⁻¹ | a⁴ + 2a²z² |
| 6₁ | 2t - t² - t⁻¹ + t⁻² - t⁻³ | 2t - 3 + 2t⁻¹ | a⁴ - 2a² + 2 |
| 5₁ | t² + t⁴ + t⁶ - t⁷ - t⁸ | t² - t + 1 - t⁻¹ + t⁻² | a⁴ + a⁴z⁴ |
Knots 5₂ and 6₁ have the same Alexander polynomial but different Jones polynomials:
Jones polynomial can distinguish them!
The HOMFLY polynomial is a 2-variable generalization that contains both Jones and Alexander:
It's the most powerful of the three!
Different polynomial invariants have different discriminating power:
Polynomial invariants are more powerful than simple numeric invariants. This demo shows examples where polynomial invariants succeed in distinguishing knots where simpler invariants fail.
Crossing number and signature distinguish these, but Alexander polynomial fails. Jones succeeds!
This demo shows the progressive power of knot invariants:
J.W. Alexander introduces the first polynomial invariant, derived from the homology of cyclic covers. Revolutionary for its time but cannot detect chirality.
Vaughan Jones discovers the Jones polynomial while studying von Neumann algebras and operator theory. It can detect chirality and distinguish knots the Alexander polynomial cannot. This breakthrough earned Jones the Fields Medal in 1990.
Six mathematicians (Hoste, Ocneanu, Millett, Freyd, Lickorish, Yetter) independently discover a 2-variable generalization that contains both Jones and Alexander as special cases. The most powerful of the classical polynomials!
Khovanov homology and other categorifications provide even deeper invariants. Connections to quantum field theory, topological quantum computing, and string theory continue to emerge.
The Jones polynomial emerges from the Yang-Baxter equation in statistical mechanics. Knot diagrams correspond to partition functions of 2D lattice models!
Edward Witten showed the Jones polynomial arises from Chern-Simons theory, a topological quantum field theory. This earned him a Fields Medal in 1990!
The Jones polynomial is connected to topological quantum computing via anyons and braid groups. Computing it efficiently is BQP-complete!
DNA strands can form knots! Polynomial invariants help biologists understand DNA recombination and the action of topoisomerase enzymes.
1. Jones Polynomial: We explored the revolutionary Jones polynomial V(t), which can detect chirality and distinguish many knots. The right-handed trefoil has V(t) = t + t³ - t⁴, while its mirror has V(t) = t⁻¹ + t⁻³ - t⁻⁴. This discovery earned Vaughan Jones the Fields Medal!
2. Alexander Polynomial: We studied the classical Alexander polynomial Δ(t), which is always symmetric but easier to compute. While it cannot detect chirality, it remains a valuable tool. Notably, knots 5₂ and 6₁ have the same Alexander polynomial but different Jones polynomials!
3. Polynomial Comparison: We compared Jones, Alexander, and HOMFLY polynomials side-by-side. HOMFLY is a 2-variable generalization that contains both Jones and Alexander as special cases, making it the most powerful classical polynomial invariant.
4. The Power of Polynomials: We saw concrete examples where polynomial invariants succeed in distinguishing knots that simpler invariants (crossing number, signature, tricolorability) cannot. This demonstrates why polynomial invariants are essential tools in modern knot theory.
Next Up: Now that you understand polynomial invariants, you're ready to explore the fundamental group - an algebraic invariant that captures even deeper topological information! The fundamental group connects knot theory to the Group Theory module you may have already studied.