P-polynomial

The set of all polynomials over J g which satisfies the property that any two polynomials, fi x) and f2 x), are equal if and only if fi x) — f2 x) = aa x), where a JFq, constitutes a ring called the ring of polynomials over J-g modulo a x). The ring of polynomials over J g modulo p x), where p x) is an irreducible polynomial, is also a field. If d p] = k, then this field is represented by the set of all p polynomials of degrees fc—1 or less over J g and is called the Galois field of order p. Every finite field J-g is isomorphic (be. can be put into a one-to-one correspondence) with some... 

The P-polynomial is defined recursively. This means that we compute the P-polynomial of an oriented link in terms of the polynomials of simpler oriented links, which in turn are computed in terms of the polynomials of oriented links which are simpler still, and so on until we get a collection of unknots each of whose polynomial is known to equal 1. 

The P-polynomial P(L) has variables m and / and is formally defined from the oriented projection of L by using the following two axioms. 

From its definition, the P-polynomial appears to depend on the particular projection of the link which we are working with. However, when this polynomial was defined it was proven that given any oriented link, no matter how it is deformed or projected, the link will always have the same P-polynomial [5, 6]. This means that two oriented links which are topologically equivalent have the same P-polynomial. In particular, if an oriented link can be deformed to its minor image then it and its minor image will have the same P-polynomial. 

We illustrate how to compute the P-polynomial by evaluating a couple of examples. First let Lq consist of the oriented unlink of two components which is illustrated in Figure 4. Then L+ and L are as shown in the figure. Now we use the second axiom of the definition of the P-polynomial, together with the observation that both L+ and L are topologically equivalent to the unknot, in order to obtain the equation / + Z1 + mP(Lo) = 0. Hence, we conclude that P(Lo) = -m l + I"1). 

We shall use this result to enable us to compute the P-polynomial of the oriented link, illustrated as L in Figure 5. This link is known as the Hopf link. We choose the upper crossing to change so that we have L+ and Lq as indicated in the figure. 

It can be seen from these simple examples that computing the P-polynomial of any complicated oriented link will be quite cumbersome. However, there are a number of excellent computer programs which will compute all of the link polynomials for any oriented link which can be drawn with up to about 50 crossings (for example, the program KNOTTHEORY by COMPUTER written by M. Ochiai, and available through anonymous ftp [7]). 

Figure 5. We compute the P-polynomial of the oriented Hopf link which is represented by L. ...

In order to see how to apply this theorem we can consider the oriented Hopf link which was illustrated in Figure 5. We determined above that the P-polynomial of this oriented Hopf link L is P(L) = Z3m 1 + lm l - Im. If we interchange l and r1 we will obtain the polynomial P(JL) = / 3m-1 + - rlm. Since P(L) ... 

Note that the theorem does not detect all topologically chiral knots and oriented links, because there are topologically chiral knots and oriented links whose P-polynomials are nonetheless symmetric with respect to / and l"1. For example, consider the knot which is illustrated in Figure 11. This knot is known by knot theorists as 942 because this is the forty second knot with 9 crossings listed in the standard knot tables (see the tables in Rolfsen s book [9]). Using a computer program we find that the P-polynomial of the knot 942 is P(942) = (-21 2 - 3-212) + m2 l 2 + 4 + l2) - m. Observe that this polynomial is symmetric with respect... 

We can also use link polynomials to prove that certain unoriented links are topologically chiral. For example, let L denote the (4,2)-torus link which is illustrated on the left in Figure 12. This is called a torus link because it can be embedded on a torus (i.e. the surface of a doughnut) without any self-intersections. It is a (4,2)-torus link, because, when it lies on the torus, it twists four times around the torus in one direction, while wrapping two times around the torus the other way. Let L denote the oriented link that we get by putting an arbitrary orientation on each component of the (4,2)-torus link, for example, as we have done in Figure 12. Now the P-polynomial of L is P(L ) = r5m l - r3m x + ml 5 -m3r + 3m r3. 

By setting in (4.1.20) A = 0, one sees that the roots of the p-polynomial determine zero stability ... 

Thus we have demonstrated how the L j /(p) polynomials can be generated and that they do satisfy the general associated Laguerre polynomial equation. Schrodinger worked out the Hydrogen orbitals from these functions in his third revolutionary paper [7] and perhaps we can appreciate the patience required to carry the derivation to useful results ... 

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