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,
- for all x
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
The example is a working of the cinquefoil knot. For convenience we'll let A=x-1/2-x1/2. Patch one of its crossings so:
- P(image:skein-relation-cinquefoil-minus-sm.png) = A×P(image:skein-relation-cinquefoil-zero-sm.png) + P(image:skein-relation-cinquefoil-plus-sm.png)
- P(image:skein-relation-link24-minus-sm.png) = A×P(image:skein-relation-link24-zero-sm.png) + P(image:skein-relation-link24-plus-sm.png)
- P(image:skein-relation-trefoil-minus-sm.png) = A×P(image:skein-relation-trefoil-zero-sm.png) + P(image:skein-relation-trefoil-plus-sm.png)
- P(image:skein-relation-link22-minus-sm.png) = A×P(image:skein-relation-link22-zero-sm.png) + P(image:skein-relation-link22-plus-sm.png)
- P(image:skein-relation-link20-minus-sm.png) = A×P(image:skein-relation-link20-zero-sm.png) + P(image:skein-relation-link20-plus-sm.png)
| knot name | diagram(s) | P(diagram) | ||
|---|---|---|---|---|
| eq'n | abbr'd | in full | ||
| unknot | image:skein-relation-link20-minus-sm.png image:skein-relation-link20-plus-sm.png image:skein-relation-link22-zero-sm.png image:skein-relation-trefoil-minus-sm.png | 1 | x→1 | |
| image:skein-relation-link20-zero-sm.png image:skein-relation-link22-minus-sm.png | 1=A?+1 | 0 | x→0 | |
| (Hopf link)[1] | image:skein-relation-link22-plus-sm.png image:skein-relation-link24-minus-sm.png image:skein-relation-trefoil-zero-sm.png | 0=A1+? | -A | x→x1/2-x-1/2 |
| trefoil | image:skein-relation-link24-zero-sm.png image:skein-relation-trefoil-plus-sm.png image:skein-relation-cinquefoil-minus-sm.png | 1=A(-A)+? | 1+A2 | x→x-1-1+x |
| image:skein-relation-link24-plus-sm.png image:skein-relation-cinquefoil-zero-sm.png | -A=A(1+A2)+? | -A(2+A2) | x→-x-3/2+x-1/2-x1/2+x3/2 | |
| cinquefoil | image:skein-relation-cinquefoil-plus-sm.png | 1+A2=A(-A(2+A2))+? | 1+3A2+A4 | x→x-2-x1+1-x+x2 |
Hints:
- A=(1-x)/x1/2
- A2=(1-2x+x2)/x
- A3=(1-x)3/x3/2=(1-3x+3x2-x3)/x3/2
- A4=(1-x)4/x2=(1-4x+6x2-4x3+x4)/x2
