# Can we get the HOMFLY polynomial for a torus knot from the Kauffman Polynomial?

This is essentially a yes/no/reference request question. I posted it on math.se and left it there for 5 days before posting here.

Let me first just ask my question: Is there a known relationship between the HOMFLY and Kauffman polynomials of torus knots? In particular, is there a published result which says that given the Kauffman polynomial of a torus knot, I can compute its HOMFLY polynomial? If so, any reference would be appreciated.

Here is is some more info:

In the paper http://arxiv.org/abs/q-alg/9507031 of Labastida and Perez, the authors claim that the HOMFLY polynomial of a torus knot can be obtained from the Kauffman Polynomial of the knot. (For the curious: The authors compute a formula for the Kauffman Polynomial of the (m,n) Torus knot which has a shape very similar to a formula for the HOMFLY polynomial discovered (I believe) by Jones. They then note how the HOMFLY formula can be obtained from their Kauffman formula. This relationship is on p. 35 of the paper linked above.)

My concern is that the derivation of formula for the Kauffman polynomial is physics and not mathematics. If anyone is familiar with the above paper and wants to tell me that it is math, I would be happy to hear it. I am not asking anyone to do my work for me and read the paper.

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## 1 Answer

John -- a formal answer is yes: The 2-variable Kauffman polynomial of a torus $(m,n)$-knot determines the Jones polynomial, which allows one to determine the $m$ and $n$ from the formula

$$V(K_{m,n})=\frac{t^{{(m-1)(n-1)}/2}(1-t^{m+1}-t^{n+1}+t^{m+n})}{1-t^2},$$

which allows one to compute the HOMFLY polynomial. Now, this is not really an answer, as you are presumably interested in how to find and rigorously prove an explicit formula for the Kauffman and Homfy polynomials for torus knots or links. I'm sure this has been done but can't think of a reference. If I find one, I'll edit my posting to include it.

A small remark: the HOMFLY polynomial wasn't discovered by Jones. It was discovered by the 6 authors the first letters of whose names make up HOMFLY, and also 2 other the first letters of whose names don't: Hoste, Ocneanu, Millett, Freyd, Lickorish, Yetter, Przytycki and Traczyk. They were certainly inspired by Jones's work though.

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I think the OP meant that the formula was discovered by Jones.... –  Ben Webster Nov 17 '12 at 13:08
Ben -- yes, that's possible; the phrasing is a bit ambiguous. –  algori Nov 17 '12 at 15:57