A knot polynomial is a polynomial whose coefficients encode some of the properties of a given knot. The purpose of such constructions is to associate to each knot a particular knot invariant which, by definition, will yield the same invariant for any two "equivalent" knots. Generally, such a polynomial is not meant to be evaluated as a function, but instead is used as a means of differentiating between knots that are not equivalent. In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... Some knots: 1. ... A knot invariant is a useful tool in knot theory. ... In general, a function is part of an answer to a question about why some object or process occurred in a system that evolved or was designed with some goal. ...

Reasoning

Many complexities of mathematical problems are made easier by reducing to previously familiar problems. In knot theory, different methods are used to match a knot to another mathematical property which is easier to use. One method is the use of polynomials.

A polynomial representation of a knot is a mapping of the mathematical properties of knots to those of polynomials. This is done because the mathematical representation of polynomials is much easier to manipulate than that of knots. The polynomial notation of a knot is also more succinct, and is much easier than, for example, drawing all the complexities of a knot. It is also easier to compare different properties of knots (such as equivalence) by using only polynomials. If the knot-to-polynomial mapping can be calculated from elements of the knot and is sufficiently discriminating (that is, the mapping can tell apart lots of different varieties of knots), two complicated knots can be checked for equivalence 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] 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. Skein relations are a piece of knot theory usually used to recursively define knot polynomials using knot diagrams as bookkeeping (compare Stückelberg-Feynman diagrams). ...

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. J. W. Alexander James Waddell Alexander II (September 19, 1888 – September 23, 1971) was an important topologist of the pre-WWII era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others. ... 1923 was a common year starting on Monday (link will take you to calendar). ... 1928 was a leap year starting on Sunday (link will take you to calendar). ... Generator redirects here. ... In Ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R. More specifically: a left principal ideal of R is a subset of R of the form Ra := {ra : r in R... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers like even number or multiple of 3. For instance, in rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalization of... Homology is an important concept in several disciplines: Homology (anthropology) in archaeology and anthropology. ... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X... In mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a solid torus into the 3-sphere. ...

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. For the square matrix section, see square matrix. ... In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

Example

On a trefoil knot:

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

resulting in the matrix

Image File history File links trefoil knot, directed, crossings numbered File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

Take the minor M₂₃

In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. ...

trefoil: x² − x + 1

Image File history File links left-handed trefoil knot, 64x64 pixels aka (3,2)-torus knot File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links right-handed trefoil knot, 64x64 pixels aka (3,2)-torus knot File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

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

Image File history File links Stevedore knot, numbered and directed, 128 pixel high File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

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. Image File history File links Figure-8 knot, 64x64 pixel File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

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. Image File history File links Granny knot, 64 pixels high File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Square knot, 64 pixels high File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

See skein relations for a second way to compute Alexander polynomials. Skein relations are a piece of knot theory usually used to recursively define knot polynomials using knot diagrams as bookkeeping (compare Stückelberg-Feynman diagrams). ...

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. John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... Skein relations are a piece of knot theory usually used to recursively define knot polynomials using knot diagrams as bookkeeping (compare Stückelberg-Feynman diagrams). ...

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

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.) 1984 is a leap year starting on Sunday of the Gregorian calendar. ... Vaughan Frederick Randal Jones (born 31 December 1952) is a New Zealand mathematician, known for his work on von Neumann algebras, knot polynomials and conformal field theory. ... A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space which is closed in the weak operator topology, or equivalently, in the strong operator topology (under pointwise convergence) and contains the identity operator. ... Fig. ...

In skein relation Skein relations are a piece of knot theory usually used to recursively define knot polynomials using knot diagrams as bookkeeping (compare Stückelberg-Feynman diagrams). ...

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

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. 1985 is a common year starting on Tuesday of the Gregorian calendar. ... Dr. Józef Henryk Przytycki is a mathematician specializing in the fields of knot theory and topology. ...

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: Skein relations are a piece of knot theory usually used to recursively define knot polynomials using knot diagrams as bookkeeping (compare Stückelberg-Feynman diagrams). ...

(Doll & Hoste 1991, Kanenobu & Sumi 1993)

(Kauffman 1991)

(Lickorish & Millett 1988)

For maximal confusion there is also a ternary form 1991 is a common year starting on Tuesday of the Gregorian calendar. ... 1993 is a common year starting on Friday of the Gregorian calendar and marked the Beginning of the International Decade to Combat Racism and Racial Discrimination (1993-2003). ... 1991 is a common year starting on Tuesday of the Gregorian calendar. ... 1988 is a leap year starting on a Friday of the Gregorian calendar. ...

For a link L of n unlinked unknots, a common thing in skein recurrences, it is easily shown (by induction) that Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. ...

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 Image::Knot-cinquefoil-sm.png 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 Image:Knot-10-56-sm.png, are in the same boat; thus it is not helpful to pair polynomial and crossings.) Image File history File links Mathworlds 10-25 knot (redrawn), small File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

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

1988 is a leap year starting on a Friday of the Gregorian calendar. ...

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

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

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

Unworked examples

knot K	Alexander ΔK(x)	Conway	Jones VK(x)
unknot 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

Image File history File links left-handed trefoil knot, 64x64 pixels aka (3,2)-torus knot File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links right-handed trefoil knot, 64x64 pixels aka (3,2)-torus knot File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links cinquefoil knot, small aka Solomons seal, double overhand, and 51. ... Image File history File links Figure-8 knot, 64x64 pixel File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Square knot, 64 pixels high File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Granny knot, 64 pixels high File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links stevedore knot, small aka 61 Mathworld · Knot Atlas · Knot Server File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

(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!