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).

The diagrams have to be directed and slightly generalised, representing multiple knots instead of just one. Pedants refer to them as link diagrams.

Given three link diagrams that are identical except for one crossing, the three are labelled as follows. Turn the diagrams so the directions at that spot are both roughly northward. One diagram will have northwest over northeast, it is labelled L_-. Another will have northeast over northwest, it's L₊. The remaining diagram is lacking that crossing and is labelled L₀.

(The labelling is actually independent of direction insofar as it remains the same if all directions are reversed. Thus polynomials on undirected knots are unambiguously defined by this method. However, the directions on links are a vital detail to retain as one recurses through a polynomial calculation.)

It is also sensible to think in a generative sense, by taking an existing link diagram and "patching" it to make the other two—just so long as the patches are applied with compatible directions.

To recursively define a knot (link) polynomial, a function F is fixed and for any triple of diagrams and their polynomials labelled as above,

or more pedantically

for all x

(Finding an F which produces polynomials independent of the sequences of crossings used in a recursion is no trivial exercise.)

Example

Sometime in the early '60s, Conway showed how to find Alexander polynomials using skein relations. As a recursion, it's not quite so direct as the matrix method; on the other hand, parts of the work done for one knot will apply to others. In particular, the network of diagrams is the same for all skein-related polynomials.

Let function P from diagrams to Laurent series in be such that and a triple of skein-relation diagrams satisfies the equation

Then P maps a knot to one of its Alexander polynomials.

The example is a working of the cinquefoil knot. For convenience we'll let A=x^-1/2-x^1/2. Patch one of its crossings so:

P() = A×P() + P()

The first diagram is actually a trefoil; the second diagram is two unknots with four crossings. Patching the latter

P() = A×P() + P()

gives, again, a trefoil, and two unknots with two crossings. Patching the trefoil

P() = A×P() + P()

gives that 2-crossing link and the unknot. Patching that link

P() = A×P() + P()

gives a link with 0 crossings. That takes a bit of sneakiness:

P() = A×P() + P()

We now have enough relations to compute the polynomials:

knot name	diagram(s)	P(diagram)
		eq'n	abbr'd	in full
unknot			1	x→1
		1=A?+1	0	x→0
(Hopf link)[1]		0=A1+?	-A	x→x1/2-x-1/2
trefoil		1=A(-A)+?	1+A2	x→x-1-1+x
		-A=A(1+A2)+?	-A(2+A2)	x→-x-3/2+x-1/2-x1/2+x3/2
cinquefoil		1+A2=A(-A(2+A2))+?	1+3A2+A4	x→x-2-x1+1-x+x2

Hints:

A=(1-x)/x^1/2

A²=(1-2x+x²)/x

A³=(1-x)³/x^3/2=(1-3x+3x²-x³)/x^3/2

A⁴=(1-x)⁴/x²=(1-4x+6x²-4x³+x⁴)/x²

External links

• AMS
• Mathworld
• HOMFLY polynomial of decorated Hopf link