"Polynomial" is used in a much more general sense than is usual. As functions in x, these are actually Laurent polynomials in x1/n for various n.
Why bother? For one thing, a polynomial is much easier to communicate than a knot, or even a drawing of a knot.
For another, it's far easier to compare two polynomials for equivalence than two knots. If the knot-to-polynomial mapping can be calculated from elements of the knot and is sufficiently discriminating, two complicated knots can be checked for identity algorithmically.
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 0.264×1.658E 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 generator 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:
Justification
Alexander polynomial
The result is ‘the’ Alexander polynomial of the knot.
knot | crossings | ||
---|---|---|---|
n | p | q | |
1 | 2 | 3 | |
2 | 3 | 1 | |
3 | 1 | 2 |
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 |
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 |
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 xn and collect xi and x-i terms. Eg, trefoil: figure-eight: granny/square: stevedore:
Alexander-Conway polynomial
Even before Conway found the skein-relation approach to the Alexander polynomials, a second form via change of variable was apparent. But Conway gets the credit.
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
Jones polynomial
In 1984 Vaughn F. R. Jones came out with the first really new knot polynomial since Alexander's. He was tinkering in his specialty, von Neumann algebras, and almost by accident found this linkage to knot theory. (Knot theory began with an idea that atoms were knotted æther vortices, and von Neumann algebras are key to quantum theory, the successor to atomic study. Jones' discovery was thus a sort of family reunion.)
Can sometimes distinguish a knot from its reflection; this is the great "breakthrough" over the Alexander and Conway polynomials.
HOMFLY(PT) polynomial
Jones' discovery prompted a hunt for a structure above his polynomial and Alexander's. Five collaborations found one essentially simultaneously; four published jointly in 1985 rather than fight over priority. "HOMFLY" is derived from their initials: Jim Hoste, Adrian Ocneanu, Kenneth C. Millett, Peter J. Freyd, W. B. Raymond Lickorish, and David N. Yetter. Some authors write "HOMFLYPT" to include the pair of Poles, Józef H. Przytycki and Pawel Traczyk, who got left out due to slow mail service.
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 form
Such 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.>
It is a generalisation of the Jones polynomial
It relates to Kauffman's unary polynomial as
BLM/Ho polynomial
Kauffman unary polynomial
Louis H. Kauffman has two knot polynomials to his credit.
Also known as normalised bracket polynomial. Denoted by by Kauffman but other authors have used different letters. It is very like the Jones polynomial:
Kauffman binary polynomial
but other than having more terms than the HOMFLYPT polynomial, its relation to the latter is unknown.
knot K | Alexander | Conway | Jones |
---|---|---|---|
unknot | 1 | 1 | 1 |
left trefoil | |||
right trefoil | |||
(right?) cinquefoil | |||
figure-8 | |||
square | |||
(left?) granny | |||
stevedore |
(Composing notes)