A knot polynomial is a particular knot invariant. The coefficients are the important part; the polynomial is not meant to be evaluated, but merely a way of indexing a set of numbers.

"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 n.

Justification

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 0.264×1.658^E knots—but is hard to compute.[1] (http://www.fortunecity.com/emachines/e11/86/knotprob.html) actually it looks like E increases rather rapidly, wrt to crossings, so "rather well" may be optimistic There is also the ropelength[2] (http://torus.math.uiuc.edu/jms/Papers/thick/ropelen.pdf).

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.

Alexander polynomial

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:

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:

M_nn = 1 - x

M_np = x

M_nq = - 1

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.

The result is ‘the’ Alexander polynomial of the knot.

Example

On a trefoil knot:

knot	crossings
	n	p	q
	1	2	3
	2	3	1
	3	1	2

resulting in the matrix

Take the minor M₂₃

trefoil: x² - x + 1

Example 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

to make the matrix

resulting in 2x² - 5x + 2

figure-eight: x² - 3x + 1

Suppose there is a knot and a plane which touches the knot at exactly two points (this may need stricting-up). The portion of the knot which lies on one side of the plane, closed with the segment joining the two points, is another knot. The original knot is said to be a sum of the two lesser knots so formed. A knot which can divide into naught but the unknot and itself is said to be prime.

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.)

x⁴ - 2x³ + 3x² - 2x + 1

Ref: Mark Anthony Armstrong Basic Topology (Springer-Verlag 1987) p237–9

Note: Because of the Mathworld (http://mathworld.wolfram.com/AlexanderPolynomial.html) 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: (x + x ^{- 1}) - 1 figure-eight: (x + x ^{- 1}) - 3 granny/square: (x² + x ^{- 2}) - 2(x + x ^{- 1}) + 3 stevedore: 2(x + x ^{- 1}) - 5

• this (http://164.8.13.169/Enciklopedija/math/math/a/a116.htm) seems to be a copy of Mathworld
• ditto (http://www.albanyconsort.com/polyinv/polyinv.pdf)
• second Alexander polynomial (http://www.inst.bnl.gov/~wei/oth.html)

See skein relations for a second way to compute Alexander polynomials.

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

with

It relates to the normalised Alexander polynomial Δ as

• Ref Mathworld (http://mathworld.wolfram.com/ConwayPolynomial.html)

Jones polynomial

In 1984 Vaughan 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.)

In skein relation

with V_unknot(x) = 1.

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

V_L(x) = V_Γ(x ^{- 1}) where L is the reflection of Γ.

V_K(e^{2πi / 3}) = 1 and for all knots K

V_L( - 1) = Δ_L( - 1) for all links L

• Ref Mathworld (http://mathworld.wolfram.com/JonesPolynomial.html)

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 P_unknot(x,y) = 1 as with the predecessors. But three different skein relations (and thus three slightly different polynomials) are seen in the wild:

(Doll & Hoste 1991, Kanenobu & Sumi 1993)

(Kauffman 1991)

(Lickorish & Millett 1988)

For maximal confusion there is also a ternary form

For a link L of n unlinked unknots, a common thing in skein recurrences, it is easily shown (by induction) that

The simplicity of the ternary HOMFLYPT is deceptive; it actually encapsulates a significant class of knot functions. Given any three functions Q, R, S (over the same set into a field), the skein-relation equation

is satisfied by . This obviously includes the Alexander, Conway, and Jones polynomials:

Δ_L(x) = P_L(1, - 1,x ^{- 1 / 2} - x^{1 / 2})

V_L(x) = P_L(x ^{- 1}, - x,x ^{- 1 / 2} - x^{1 / 2})

Thus, to go any further with skein relations one must avoid recurrences of the above form.

Such interrelations permit facts about HOMFLYPT to be transferred (with appropriate transformation) to its predecessors. For instance, although Missing image
:Knot-cinquefoil-sm.png
the cinquefoil

and Missing image
:Knot-10-132-sm.png
image::Knot-10-132-sm.png

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 Missing image
Knot-10-56-sm.png
image:knot-10-56-sm.png

, 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 (http://mathworld.wolfram.com/HOMFLYPolynomial.html)
• Calculating HOMFLY by Dynamic Programming (http://burtleburtle.net/bob/knot/thesis.html)

BLM/Ho polynomial

• Brandt, Lickorish, Millett, Ho
• Is an invariant of unoriented knots and links, with Q_unknot(x) = 1. It was derived as a symmetrization of the HOMFLY (PT) Polynomial, and necessarily introduced an term in the skein relation equation. Because it is independent of the orientations of the components of the link, it defines equivalence classes of point sets. (Note: This was the goal of the original derivation. - R. D. Brandt.)

• Ref Mathworld (http://mathworld.wolfram.com/BLMHoPolynomial.html)

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:

• Ref Mathworld (http://mathworld.wolfram.com/KauffmanPolynomialX.html)

Kauffman binary polynomial

It is a generalisation of the Jones polynomial

V(x) = F( - x^{3 / 4},x ^{- 1 / 4} + x^{1 / 4})

but other than having more terms than the HOMFLYPT polynomial, its relation to the latter is unknown.

It relates to Kauffman's unary polynomial as

• Ref Mathworld (http://mathworld.wolfram.com/KauffmanPolynomialF.html)

Unworked examples

knot K	Alexander ΔK(x)	Conway	Jones VK(x)
unknot Missing image Knot-unknot-64.png image:knot-unknot-64.png	1	1	1
left trefoil	(x + x - 1) - 1	1 + x2	- x - 4 + x - 3 + x - 1
right trefoil			x + x3 - x4
(right?) cinquefoil	(x2 + x - 2) - (x + x - 1) + 1	1 + 3x2 + x4	x2 + x4 - x5 + x6 - x7
figure-8	(x + x - 1) - 3	1 - x2	x - 2 - x - 1 + 1 - x + x2
square	(x2 + x - 2) - 2(x + x - 1) + 3	(1 + x2)2
(left?) granny
stevedore	2(x + x - 1) - 5	1 - 2x2	x - 2 - x - 1 + 2 - 2x + x2 - x3 + x4

(Composing notes)

• Saw mention of a Millet-Prztycki-Traczyk polynomial by three of the HOMFLYPTers...
• WTF Siefert matrices? (http://maths.dur.ac.uk/~dma0ck/img/gtopex.ps)
• [3] (http://www.inst.bnl.gov/~wei/homflypt.html)
• might be useful for something (http://www.inst.bnl.gov/~wei/example.html)
• HOMFLY this! (http://www.pims.math.ca/knotplot/Ashley/)