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.
This module covers 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 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 is a fundamental topological invariant -- no matter how you deform the knot, you cannot 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 |
Key insight: If two knots have different crossing numbers, they cannot be equivalent. However, computing the crossing number is surprisingly difficult -- for many knots, we still do not know if their minimal diagrams have been found.
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.
Key insight: Since the unknot is not tricolorable but the trefoil is, we can prove that the trefoil is knotted. Tricolorability is preserved under the Reidemeister moves, making it a true knot invariant.
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.
Key insight: 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. Computing exact unknotting numbers remains an active area of research.
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).
Key insight: The signature is always an even integer. If a knot has nonzero signature, it is chiral (not equivalent to its mirror image). The signature is also additive under connected sum, making it a powerful algebraic invariant.