Talk:Knot polynomial

Twisting number?

what with the twisting number???

As you define it, the polynomials are not invariants! Just look at the figure 8 knot, which is the unknot twisted once. You have to normalize by dividing the polynome by the Kaufman relation. Also, what of disjoint sum of links? mousomer

Comment: The figure eight knot does not look like the number eight. What you are talking about is indeed the unknot. — Preceding unsigned comment added by 69.216.247.251 (talk) 23:50, 14 November 2005

What does it mean to "drop any one row" ??? Which row should I drop?? — Preceding unsigned comment added by 217.229.131.111 (talk) 05:37, 27 February 2004

Whatever row you wish.

The 'n' rows you get from 'n' crossings are linearily dependent.

Any 'n-1' of the rows are linearily independent.

So just drop any one row+column to get a regular matrix. mousomer

Needs improvement

This article really could use some improvement. First of all, it never mentions that the important part is not that these invariants are polynomials in one variable but that each coefficient is an invariant of a very special type, called Vassiliev invariant, or finite type invariant. Almost all of them can be obtained from the theory of quantum groups; they are also related to topological quantum field theory. This is a very rapidly developing field, with a huge body of literature available (such as Bar-Natan's paper in Topology). I know this field; however, I do not feel I could rewrite this article. I would rather write a new article, on Vassiliev invariants, from scratch. --Kirillov 1 Jan 2001 kirillov at math sunysb edu

I was planning on writing something on finite type invariants sometime. --C S 21:34, 26 October 2005 (UTC)

Hmmm... this page is a disaster. Perhaps we should just start over, with some separate articles... --ScottMorrison 04:46, 5 Apr 2005 (UTC)

I agree this page is a disaster. I would like to delete the "Reasoning" section entirely, since it's mostly wrong (e.g., in the motivation for considering knot polynomials) when it's not irrelevant (e.g., in mentioning knot energies). The introduction needs to be rewritten as well, to include the relation to quantum field theory. For the Alexander polynomial, the algebraic-topological definition should be given as the primary definition. Probably the specific polynomials should get their own page. Any objections to me doing this? --Dylan Thurston 8 May 2205

No. One of the most irksome things is to get redirected to this page that is way too long. Each polynomial should definitely have its own page. --C S 21:34, 26 October 2005 (UTC)

I rewrote the introduction because I didn't like the wording and it seemed vague in its description of "encoding a sequence of numbers". The "Reasoning" section ought to be torn apart and the relevant details put in other places. A "History" section may be appropriate, either at the beginning or end of the article, depending on taste. Of course, the particular polynomials that are of importance should have their own separate articles. Perhaps we should use something like Alexander for all the basic examples (of which there should be many). I feel that this article should be an introduction and that more technical details should be pushed back into separate articles on specific invariants. - Gauge 07:59, 26 Jun 2005 (UTC)

Recently deleted comment from the article

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. E.g., trefoil: $(x+x^{-1})-1$ figure-eight: $(x+x^{-1})-3$ granny/square: $(x^{2}+x^{-2})-2(x+x^{-1})+3$ stevedore: $2(x+x^{-1})-5$

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

Nothing mysterious here. Surely this is just the connection between the Alexander polynomial and the Conway polynomial (as it pointed out further down in the main article). Gandalf61

Removed comment

I removed a comment from the text (the italic bit) from the reasoning section:

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

I don't know if we give an equation for E anywhere, so I don't know if the comment is sensible (and it is all quite a long way over my head anyway). Andreww 05:58, 16 October 2005 (UTC)

I also removed the following - this is an encyclopedia:

<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.>

Andreww 10:15, 24 October 2005 (UTC)

And some more - may be useful for future editors:

==(Composing notes)==

• Saw mention of a Millet-Prztycki-Traczyk polynomial by three of the HOMFLYPTers...
• [2]
• [3]
• might be useful for something
• HOMFLY this!

Andreww 10:19, 24 October 2005 (UTC)

Most of content from original, long, rambling messy article

Here's the stuff, if people want to make use of it. I just started a new article, which should hopefully grow in a much better fashion. Heck, hopefully it'll grow! --C S 01:08, 2 November 2005 (UTC)

It seems Kwantus was behind much of the original article, and he uploaded many figures (especially some nice ones to for the skein relation article) and did much work. Unfortunately, he seems to be deceased, which explains why he hasn't been able to help out here. Anyway, it would be a real shame if we didn't use some of his stuff. I think a good idea is to take some of the examples and work he did and incorporate it into the individual polynomial pages. Some of the skein relation stuff may need to be moved to the Alexander polynomial page, and skein relation can be more general (rather than focus on specifics). --C S 07:37, 11 November 2005 (UTC)

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. 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.

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

${\begin{pmatrix}1-x&-1&x\\x&1-x&-1\\-1&x&1-x\\\end{pmatrix}}$

Take the minor M₂₃

$M_{23}={\begin{vmatrix}1-x&-1\\-1&x\\\end{vmatrix}}=-x^{2}+x-1$

trefoil: $x^{2}-x+1$

Example 2

On a stevedore knot:

knot	crossings
	n	p	q
	1	3	6
	4	6	5
	5	3	2
	6	4	1
	3	2	1
	2	4	5

to make the matrix

${\begin{pmatrix}1-x&0&x&0&0&-1\\0&1-x&0&x&-1&0\\x&-1&1-x&0&0&0\\0&0&0&1-x&-1&x\\0&-1&x&0&1-x&0\\-1&0&0&x&0&1-x\end{pmatrix}}$

resulting in $2x^{2}-5x+2$

figure-eight: $x^{2}-3x+1$

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.)

$x^{4}-2x^{3}+3x^{2}-2x+1$

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

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 $\nabla _{L}$ for a link (generalised knot) L. Its skein-relation equation is

$\nabla _{L_{-}}(x)+x\nabla _{L_{0}}(x)=\nabla _{L_{+}}(x)$

with $\nabla _{\rm {unknot}}(x)=1$

It relates to the normalised Alexander polynomial $\Delta$ as

$\Delta _{L}(x^{2})=\nabla _{L}(x-x^{-1})$

• Ref Mathworld

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

$xV_{L_{-}}(x)+(x^{1/2}-x^{-1/2})V_{L_{0}}(x)=x^{-1}V_{L_{+}}(x)$

with $V_{\rm {unknot}}(x)=1$.

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

$V_{\rm {L}}(x)=V_{\Gamma }(x^{-1})$ where L is the reflection of $\Gamma$.

$V_{K}(e^{2\pi i/3})=1$ and $V_{K}^{\prime }(1)=0$ for all knots K

$V_{L}(-1)=\Delta _{L}(-1)$ for all links L

• Ref Mathworld

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 $P_{\rm {unknot}}(x,y)=1$ as with the predecessors. But three different skein relations (and thus three slightly different polynomials) are seen in the wild:

$xP_{L_{-}}(x,y)+yP_{L_{0}}(x,y)=x^{-1}P_{L_{+}}(x,y)$ (Doll & Hoste 1991, Kanenobu & Sumi 1993)

$x^{-1}P_{L_{-}}(x,y)+yP_{L_{0}}(x,y)=xP_{L_{+}}(x,y)$ (Kauffman 1991)

$x^{-1}P_{L_{-}}(x,y)+yP_{L_{0}}(x,y)+xP_{L_{+}}(x,y)=0$ (Lickorish & Millett 1988)

For maximal confusion there is also a ternary form

$yP_{L_{-}}(x,y,z)+zP_{L_{0}}(x,y,z)+xP_{L_{+}}(x,y,z)=0$

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

$P_{L}(x,y,z)=\left({x+y \over -z}\right)^{n-1}$

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

$Q(x)Z_{L_{-}}(x)+R(x)Z_{L_{0}}(x)+S(x)Z_{L_{+}}(x)=0$

is satisfied by $Z_{L}(x)=P_{L}{\Big (}S(x),Q(x),R(x){\Big )}$. This obviously includes the Alexander, Conway, and Jones polynomials:

$\Delta _{L}(x)=P_{L}(1,-1,x^{-1/2}-x^{1/2})$

$\nabla _{L}(x)=P_{L}(1,-1,-x)$

$V_{L}(x)=P_{L}(x^{-1},-x,x^{-1/2}-x^{1/2})$

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 [[image::Knot-cinquefoil-sm.png|the cinquefoil]] and [[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 File:Knot-10-56-sm.png, are in the same boat; thus it is not helpful to pair polynomial and crossings.)

Also, $P_{A\#B}(x)=P_{A}(x)P_{B}(x)$ for all knot sums $A\#B$—and the other polynomials inherit this property.

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

BLM/Ho polynomial

• Brandt, Lickorish, Millett, Ho
• Is an invariant of unoriented knots and links, with $Q_{\rm {unknot}}(x)=1$. It was derived as a symmetrization of the HOMFLY (PT) Polynomial, and necessarily introduced an $L_{\infty }$ 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.)

$Q_{L_{+}}(x)+Q_{L_{-}}(x)=x(Q_{L_{0}}(x)+Q_{L_{\infty }}(x))$

• Ref Mathworld

Kauffman unary polynomial

See main article bracket polynomial

Louis H. Kauffman has two knot polynomials to his credit. Also known as normalised bracket polynomial. Denoted by ${\mathcal {L}}$ by Kauffman but other authors have used different letters. It is very like the Jones polynomial:

${\mathcal {L}}(x)=V(x^{-4})$

• Ref Mathworld

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

${\mathcal {L}}(x)=F(-x^{-3},x+x^{-1})$

• Ref Mathworld

Unworked examples

knot K	Alexander $\Delta _{K}(x)$	Conway $\nabla _{K}(x)$	Jones $V_{K}(x)$
unknot File:Knot-unknot-64.png	1	1	1
left trefoil [[image:knot-trefoil-left.svg|65px|