Skein relations are often used to give a simple definition of knot polynomials.
Depending on the knot polynomial in question, the links (or tangles) appearing in a skein relation may be oriented or unoriented.
Turn the three link diagram so the directions at the crossing in question are both roughly northward.
However, the directions on links are a vital detail to retain as one recurses through a polynomial calculation.)
More formally, a skein relation can be thought of as defining the kernel of a quotient map from the planar algebra of tangles.
Sometime in the early 1960s, Conway showed how to compute the Alexander polynomial using skein relations.
As it is recursive, it is not quite so direct as Alexander's original matrix method; on the other hand, parts of the work done for one knot will apply to others.
At each stage we exhibit a relationship involving a more complex link and two simpler diagrams.
The unlink takes a bit of sneakiness: We now have enough relations to compute the polynomials of all the links we've encountered, and can use the above equations in reverse order to work up to the cinquefoil knot itself.
denotes the unknown quantity we are solving for in each relation: Thus the Alexander polynomial for a cinquefoil is P(x) = x−2 -x−1 +1 -x +x2.
In knot theory, the term skein appears to have been coined by John Conway around 1979, and refers to the unit of measure of yarn in the textiles industry.