The HOMFLY Polynomial

Historical Background

The publication of the Jones Polynomial excited the mathematical community to the point that new polynomial invariants were being created at a stupendous rate. One of the objectives of the time was to find a polynomial that generalized both the Alexander and Jones polynomial. The Oriented Polynomial, or HOMFLY Polynomial was a successful solution, published by several groups of mathematicians simultaneously. The paper was published under the names of Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter. The HOMFLY Polynomial uses a skein relation, like the Jones Polynomial, but the new polynomial uses two variables, unlike the Alexander and Jones Polynomials.

Setting Up

The HOMFLY Polynomial is derived from an oriented knot diagram like the one below.

Also, the HOMFLY Polynomial is calculated using a skein relation like the Jones polynomial. This relation is written as:

HP1: P(L) is an isotopy invariant
HP2: P(U) = 1
HP3: l · P(L+) + l-1 · P(L-) + m · P(L0) = 0

with the same crossing types as the Jones polynomial.

Performing the Calculations

To calculate the polynomial, a crossing is selected and the skein relation is solved for the polynomial given that type of crossing. In the example below, a crossing of type L- is selected and the relation is solved for P(L-).

P(L-) = -l · (l · P(L+) + m · P(L0))

Then the crossing is changed to type L+ and the invariant is calculated.

Due to rule HP1, this knot diagram can be simplified. A type II Reidemeister move followed by a type I Reidemeister move will produce the unknot. Rule HP2 states that the polynomial of the unknot is 1, so the value of P(L+) is 1.

P(L+) = 1

In the case of L0, some simplification of the diagram leads to the Hopf Link, and it takes some work to calculate its invariant.

P(L0) = -lm + l3m-1 + lm-1

Substituting the results into the original skein relation results in the HOMFLY Polynomial for the left-handed trefoil knot.

P(L) = l2m2 - 2 · l2 - l4

Conclusion

The HOMFLY Polynomial serves as a more general polynomial invariant that covers the Alexander and Jones Polynomials.