Crossing number, tricolorability, and unknotting number
A knot invariant is a property that remains unchanged under ambient isotopy - if two knots are equivalent, they must have the same invariant values. Invariants are the primary tool for distinguishing knots and proving two knots are different.
In this module, you'll explore four fundamental knot invariants: the crossing number (minimum crossings in any diagram), tricolorability (a simple but powerful coloring property), unknotting number (minimum crossing changes to unknot), and the signature (an integer derived from the Seifert matrix). Together, these invariants give us powerful tools for understanding and classifying knots.
The crossing number is the minimum number of crossings in any diagram of a knot. It's a fundamental topological invariant - no matter how you deform the knot, you can't reduce the crossing count below this minimum.
| Crossings | Number of Knots | Examples |
|---|---|---|
| 0 | 1 | Unknot (0₁) |
| 3 | 1 | Trefoil (3₁) |
| 4 | 1 | Figure-Eight (4₁) |
| 5 | 2 | 5₁ (Cinquefoil), 5₂ (Three-Twist) |
| 6 | 3 | 6₁, 6₂, 6₃ |
| 7 | 7 | 7₁ through 7₇ |
The crossing number is one of the most fundamental knot invariants. If two knots have different crossing numbers, they cannot be equivalent. However, the converse isn't true - two different knots can have the same crossing number!
Computing the crossing number is surprisingly difficult. For many knots, we still don't know if their minimal diagrams have been found. This remains an active area of research in knot theory.
A knot is tricolorable if you can color each strand with one of three colors such that at each crossing, either all three colors meet or only one color is present. This simple rule creates a powerful knot invariant!
Trefoil (3₁)
The trefoil is tricolorable! Each strand can be assigned one of three colors following the tricoloring rules.
Tricolorability is a knot invariant: if a knot is tricolorable, any diagram of that knot can be tricolored. More importantly, tricolorability is preserved under the Reidemeister moves!
Since the unknot is not tricolorable but the trefoil is, we can prove that the trefoil is knotted - it cannot be unknotted! This simple test gives us a powerful way to distinguish knots.
The unknotting number is the minimum number of crossing changes needed to transform a knot into the unknot. A crossing change switches an over-crossing to an under-crossing or vice versa.
Change any one of the three crossings from over to under (or vice versa) to get the unknot.
| Knot | Crossings | Unknotting # | Note |
|---|---|---|---|
| Unknot (0₁) | 0 | 0 | Already unknotted |
| Trefoil (3₁) | 3 | 1 | One change suffices |
| Figure-Eight (4₁) | 4 | 1 | One change suffices |
| Cinquefoil (5₁) | 5 | 2 | Multiple changes needed |
| Three-Twist (5₂) | 5 | 1 | One change suffices |
| Stevedore (6₁) | 6 | 1 | One change suffices |
A crossing change takes an over-crossing and changes it to an under-crossing, or vice versa. This is like cutting the rope, flipping one strand, and reconnecting it.
The unknotting number is notoriously difficult to compute! For many knots, we still don't know their exact unknotting number. It's an active area of research.
Notice that the unknotting number can be much smaller than the crossing number. The stevedore knot has 6 crossings but unknotting number 1 - you only need to change one crossing to unknot it!
The signature is an integer-valued knot invariant derived from the Seifert matrix. It can distinguish knots and detect chirality - mirror images have opposite signatures!
| Knot | Signature (σ) | Chirality |
|---|---|---|
| Unknot (0₁) | 0 | Achiral |
| Trefoil (Right-handed) (3₁) | -2 | Chiral |
| Trefoil (Left-handed) (3₁*) | +2 | Chiral |
| Figure-Eight (4₁) | 0 | Achiral |
| Cinquefoil (5₁) | -4 | Chiral |
| Three-Twist (5₂) | +2 | Chiral |
If a knot K has signature σ(K), then its mirror image K* has signature σ(K*) = -σ(K).
The signature is computed from the Seifert matrix, which is derived from a Seifert surface spanning the knot. It's always an even integer.
Key properties:
1. Crossing Number: We learned that the crossing number is the minimum number of crossings in any diagram. There are only 2 knots with 5 crossings, but 7 knots with 7 crossings - the number grows quickly!
2. Tricolorability: We discovered a simple but powerful test: the trefoil is tricolorable but the unknot is not, proving they're different! This invariant is preserved under Reidemeister moves.
3. Unknotting Number: We explored how many crossing changes are needed to unknot a knot. The trefoil needs 1 change, but the cinquefoil needs 2. This invariant is notoriously difficult to compute!
4. Signature: We calculated the signature for various knots. Mirror images have opposite signatures, so we can detect chirality. The trefoil has σ = -2, while its mirror has σ = +2.
Next Up: Now that you understand basic invariants, you're ready to explore the crown jewel of knot theory - polynomial invariants! The Jones polynomial revolutionized knot theory in the 1980s, and you'll learn how to compute it and understand what it tells us about knots.