"Polynomial" is used in a much more general sense than is usual. As functions in `x`, these are actually Laurent polynomials in `x ^{1/n}` for various

The latter condition is the harder to satisfy.

Of course polynomials are not the only things available; another hash on a knot is the least number of crossings needed in a diagram of it. But that does not discriminate knots at all well. Another hash is the Fukuhara/O'Hara energy, which discriminate fairly well—an energy `E` corresponds to at most 0.264×1.658^{E} knots—but is hard to compute.[1] *actually it looks like E increases rather rapidly, wrt to crossings, so "rather well" may be optimistic* There is also the ropelength[1].

It's also possible that elementary polynomial operations could turn out to have analogues in knot manipulations. Indeed, this is the idea behind skein relations.

James W. Alexander invented the first useful knot polynomial in 1923, and published in 1928. Technically, an Alexander polynomial is a *generato*r of a *principal Alexander ideal* related to the *homology* of the *infinitely cyclic cover* of a knot *complement*—where all the *emphasised* phrases have particular mathematical meanings. Fortunately there is a shortcut that computes the polynomial from the crossings of an oriented knot.

Procedure, somewhat informally:

- 1) Number the knot's crossings, 1…
`N`. Prepare an`N×N`matrix`M`.*(Q: does any ol' diagram do, or does it have to have minimal crossings?)* - 2) Walk along the knot. As you pass
*over*crossing`n`, with crossing`p`on the left and crossing`q`on the right, add to the matrix: - 3) Fill the rest of
`M`with zeros. - 4) Drop from
`M`any one row and any one column. - 5) Take the determinant of
`M`(this is*an*Alexander polynomial of the knot). - 6) Normalise by dropping all the zero roots and, if the highest-degree coefficient is negative, negating.

knot | crossings | ||
---|---|---|---|

n | p | q
| |

1 | 2 | 3 | |

2 | 3 | 1 | |

3 | 1 | 2 |

- resulting in the matrix
- Take the minor M
_{23}

- trefoil:

On a stevedore knot:

knot | crossings | ||
---|---|---|---|

n | p | q
| |

1 | 3 | 6 | |

4 | 6 | 5 | |

5 | 3 | 2 | |

6 | 4 | 1 | |

3 | 1 | 2 | |

2 | 4 | 5 |

- to make the matrix

1-x | 0 | x | 0 | 0 | -1 |

0 | 1-x | 0 | x | -1 | 0 |

x | -1 | 1-x | 0 | 0 | 0 |

0 | 0 | 0 | 1-x | -1 | x |

0 | -1 | x | 0 | 1-x | 0 |

-1 | 0 | 0 | x | 0 | 1-x |

- resulting in
- figure-eight:

The product of the Alexander polynomials of two knots is an Alexander polynomial of their sum. Seeing that the granny knot is the sum of two trefoils of the same hand, and the square knot is the sum of two trefoils of opposite hand, we can easily calculate their polynomial. (They share a polynomial since the handedness of a trefoil is not detected.)

*Note: Because of the **Mathworld form, I suspect Alexander polynomials have a coefficient symmetry which leads to a second canonic form. The polynomial above will have degree 2n; divide by x^{n} and collect x^{i} and x^{-i} terms. Eg, trefoil: figure-eight: granny/square: stevedore: *

- this seems to be a copy of Mathworld
- ditto
- second Alexander polynomial

This other polynomial is usually denoted for a link (generalised knot) `L`. Its skein-relation equation is

It relates to the normalised Alexander polynomial as

- Ref Mathworld

Can sometimes distinguish a knot from its reflection; this is the great "breakthrough" over the Alexander and Conway polynomials.

- where L is the reflection of .

- and for all knots
`K` - for all links
`L`

- Ref Mathworld

HOMFLYPT is a binary (two-variable) polynomial, with as with the predecessors. But three different skein relations (and thus three slightly different polynomials) are seen in the wild:

For maximal confusion there is also a ternary formSuch interrelations permit facts about HOMFLYPT to be transferred (with appropriate transformation) to its predecessors. For instance, although and are known to be different knots, their HOMFLYPTs are the same; thus they also share their Alexander, Conway, and Jones. (Worse, two 10-crossing knots, and , are in the same boat; thus it is not helpful to pair polynomial and crossings.)

Also, for all knot sums —and the other polynomials inherit this property.

*<The author is astounded that the ternary HOMFLYPT, which seems an absurdly obvious skein relation, should have lain unseen in plain sight for over 20 years. Conway must really be wondering why he didn't see it. Perhaps he thought it was too obvious to work.>*

*<The author is also puzzled that Mathworld mentions the ternary on the HOMFLYPT page as if it were a HOMFLYPT, but without specific citation, and doesn't use the form anywhere else—very odd, given that it's the form from which six other polynomials are readily found.>*

- Ivars Peterson Mathematical Tourist (1988) p70–80
- Mathworld
- Calculating HOMFLY by Dynamic Programming

- Brandt, Lickorish, Millett, Ho
- ''involves a skein rel eqn with an ... WTF? I mean, there's an obvious 4th way of patching, but not with directions...is it undirected? At first glance BLM/Ho's skein recurrence is properly symmetric for it to work... And
- Ref Mathworld

It is a generalisation of the Jones polynomial

It relates to Kauffman's unary polynomial as

knot K
| Alexander | Conway | Jones |
---|---|---|---|

unknot | 1 | 1 | 1 |

left trefoil | |||

right trefoil | |||

(right?) cinquefoil | |||

figure-8 | |||

square | |||

(left?) granny | |||

stevedore |