Skein Relation - Example

Example

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. In particular, the network of diagrams is the same for all skein-related polynomials.

Let function P from link 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.

In this example, we calculate the Alexander polynomial of the cinquefoil knot, the alternating knot with five crossings in its minimal diagram. At each stage we exhibit a relationship involving a more complex link and two simpler diagrams. Note that the more complex link is on the right in each step below except the last. For convenience, let A = x−1/2−x1/2.

To begin, we create two new diagrams by patching one of the cinquefoil's crossings (highlighted in yellow) 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 (the Hopf link ). Patching the trefoil

P = A × P + P

gives the unknot and, again, the Hopf link. Patching the Hopf link

P = A × P + P

gives a link with 0 crossings (unlink) and an unknot. The unlink takes a bit of sneakiness:

P = A × P + P

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:

Thus the Alexander polynomial for a cinquefoil is P(x) = x−2 -x -1 +1 -x +x2.

Useful formulas:

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