Knot Polynomial Justification Alexander Polynomial

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.

1 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.

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[2].

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.

2 Alexander polynomial

2.1 Example

2.2 Example 2

On a stevedore 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:

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.)

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

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:

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

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

3 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 as

Ref Mathworld

4 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.)

In skein relation

with .

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

5 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 19851985 is a common year starting on Tuesday. Events January events January 1 Creation of the Internet's Domain Name System. January 17 British Telecom annouces they are going to abolish the famous red telephone boxes. January 23 A debate in the House of Lor 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:

(Doll & Hoste 19911991 like 2002, is a palindromic year. It also has the same calendar as 2002, including Easter on March 31. It is a common year starting on Tuesday. Events January January 2 Sharon Pratt Dixon is sworn in as mayor of Washington, DC becoming the first blac, Kanenobu & Sumi 19931993 is a common year starting on Friday and marked the Beginning of the International Decade to Combat Racism and Racial Discrimination (1993-2003 Events January January 1 Czechoslovakia divides. Establishment of independent Slovakia and Czech Republic.)

(Kauffman 19911991 like 2002, is a palindromic year. It also has the same calendar as 2002, including Easter on March 31. It is a common year starting on Tuesday. Events January January 2 Sharon Pratt Dixon is sworn in as mayor of Washington, DC becoming the first blac)

(Lickorish & Millett 19881988 is a leap year starting on Friday (click on link for calendar). Events January January 2 Georgia celebrates its bicentennial statehood. January 9 Connecticut celebrates its bicentennial statehood. January 26 Australia celebrates its bicentennial day.)

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:

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 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.

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

6 BLM/Ho polynomial

Brandt, Lickorish, Millett, Ho
Is an invariant of unoriented knots and links, with . 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

7 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

8 Kauffman binary polynomial

It is a generalisation of the Jones polynomial

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

9 Unworked examples

knot `K`	Alexander	Conway	Jones
unknot	1	1	1
left trefoil
right trefoil
(right?) cinquefoil
figure-8
square
(left?) granny
stevedore

10 (Composing notes)

Saw mention of a Millet-Prztycki-Traczyk polynomial by three of the HOMFLYPTers...
WTF Siefert matrices?
[3]
might be useful for something
HOMFLY this!

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