Topological Chirality of Knots and Links

There are several link polynomials which are convenient to use, notably the Jones polynomial [3], the Kauffman polynomial [4], and the 2-variable HOMFLY polynomial [5]. These polynomials are all easy to explain and somewhat similar 

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. 

Now we see that Lq can be deformed to the unknot and L+ can be deformed to the link whose polynomial we computed previously. So we substitute these 

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