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Recurrence relation for HOMFLY polynomial and rational specializations

Turning the skein relation for HOMFLY into a Fibonacci recurrence, we prove that there are only three rational specializations of HOMFLY polynomial: Alexander-Conway, Jones, and a new one. Using the recurrence relation, we find general and relative expansion formulae and rational generating functions for Alexander-Conway polynomial and the new polynomial, which reduce the computations to closure of simple braids, a subset of square free braids; HOMFLY polynomials of these simple braids are also computed. Algebraic independence of these three polynomials is proved.