Skein Relation - Definition

Definition

A skein relationship requires three link diagrams that are identical except at one crossing. The three diagrams must exhibit the three possibilities that could occur at that crossing, it could be under, it could be over or it could not exist at all. Link diagrams must be considered because a single skein change can alter a diagram from representing a knot to representing a link and vice versa. Depending on the knot polynomial in question, the links (or tangles) appearing in a skein relation may be oriented or unoriented.

The three diagrams are labelled as follows. Turn the diagrams so the directions at the crossing in question 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

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

More formally, a skein relation can be thought of as defining the kernel of a quotient map from the planar algebra of tangles. Such a map corresponds to a knot polynomial if all closed diagrams are taken to some (polynomial) multiple of the image of the empty diagram.