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.
This module covers the Jones polynomial and the classical Alexander polynomial, compares their strengths and weaknesses, and shows 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 Laurent polynomial that can distinguish many knots that other invariants cannot. It detects chirality: if a knot K has polynomial V(t), its mirror image has V(t⁻¹).
| 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 - t + 1 - t⁻¹ + t⁻² | 1 |
| Cinquefoil (5₁) | t² + t⁴ - t⁵ + t⁶ - t⁷ | 1 |
| Three-Twist (5₂) | t - t² + 2t³ - t⁴ + t⁵ - t⁶ | 1 |
If a knot K has Jones polynomial VK(t), then its mirror image K* has polynomial VK*(t) = VK(t⁻¹).
Key insight: The Jones polynomial can detect chirality -- the right-handed trefoil has V(t) = t + t³ - t⁴, while its mirror image has V(t) = t⁻¹ + t⁻³ - t⁻⁴. These are different polynomials, proving the two trefoils are not equivalent.
The Alexander polynomial, discovered in 1928, is the classical polynomial invariant. While less powerful than the Jones polynomial, it is easier to compute and has beautiful algebraic properties, including the symmetry property that it always equals its own reciprocal.
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) | -t + 3 - t⁻¹ | 5 |
| Cinquefoil (5₁) | t² - t + 1 - t⁻¹ + t⁻² | 5 |
| Three-Twist (5₂) | 2t - 3 + 2t⁻¹ | 7 |
| Stevedore (6₁) | 2t - 5 + 2t⁻¹ | 9 |
The Alexander polynomial cannot distinguish all knots. For example, 6₁ and 9₄₆ have the same polynomial:
This is where the Jones polynomial excels - it CAN distinguish these knots!
Key insight: The Alexander polynomial cannot detect chirality (it is always symmetric), but it reveals a key limitation: knots 6₁ and 9₄₆ share the same Alexander polynomial despite being different knots. This is where the Jones polynomial excels.
Different polynomial invariants have different strengths. Compare the Jones, Alexander, and HOMFLY polynomials side-by-side to see which ones can distinguish different knots. The HOMFLY polynomial is a two-variable generalization that contains both Jones and Alexander as special cases.
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 | 2a² - a⁴ + a²z² |
| 4₁ | t² - t + 1 - t⁻¹ + t⁻² | -t + 3 - t⁻¹ | a² - 1 + a⁻² - z² |
| 5₂ | t - t² + 2t³ - t⁴ + t⁵ - t⁶ | 2t - 3 + 2t⁻¹ | a² + a⁴ - a⁶ + (a² + a⁴)z² |
| 6₁ | t⁴ - t³ + t² - 2t + 2 - t⁻¹ + t⁻² | 2t - 5 + 2t⁻¹ | a⁴ - a² + a⁻² - (a² + 1)z² |
| 5₁ | t² + t⁴ - t⁵ + t⁶ - t⁷ | t² - t + 1 - t⁻¹ + t⁻² | 3a⁴ - 2a⁶ + (4a⁴ - a⁶)z² + a⁴z⁴ |
Knots 6₁ and 9₄₆ 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!
Key insight: The HOMFLY polynomial P(a,z) generalizes both Jones and Alexander. Setting a = t, z = √t - 1/√t recovers the Jones polynomial; setting a = 1 recovers the Alexander polynomial. It is the most discriminating of the classical polynomial invariants.
Polynomial invariants are more powerful than simple numeric invariants. Compare pairs of knots to see concrete examples where polynomial invariants succeed in distinguishing knots where simpler invariants like crossing number, signature, and tricolorability fail.
Crossing number distinguishes these, but signature, tricolorability, and the Alexander polynomial all fail. Jones succeeds!
Key insight: No single invariant is perfect. Simple numeric invariants are easy to compute but limited; the Alexander polynomial was revolutionary but cannot detect chirality; the Jones polynomial distinguishes many knots Alexander cannot. We use multiple invariants together to classify knots.