This paper deals with the two variable Laurent polynomial invariant, \(P_ L(\ell,m)\) of an oriented link \(L\subset S^ 3\) often called the HOMFLY polynomial and described by \textit{P. Freyd} et. al. [Bull. Am. Math. Soc., New Ser. 12, 239-246 (1985; Zbl 0572.57002)]. Formulae are established which relate certain specific values of this polynomial to other invariants of L - namely \(P_ L(1, \sqrt{2})\) is given in terms of the Arf invariant and number of components of L, \(P_ L(1,1)\) is given in terms of the dimension of \(H_ 1(T_ L; {\mathbb{Z}}_ 2)\), and \(P_ L(e^{i\pi /6},1)\) is given in terms of the dimension of \(H_ 1(D_ L; {\mathbb{Z}}_ 2)\) and the number of components of L. Here \(D_ L\) and \(T_ L\) refer respectively to the double and triple cyclic branched cover of \(S^ 3\) branched over L.