On the HOMFLY polynomial of 4-plat presentations of knots. (English) Zbl 1347.57015

This paper gives a formula to calculate the HOMFLY polynomials of \(2\)-bridge knots by using representations of the braid group \(\mathbb{B}_4\) on \(4\)-strands into a group of \(3\times 3\) matrices.
In knot theory, the HOMFLY polynomial is a \(2\)-variable oriented link polynomial introduced in [P. Freyd et al., Bull. Am. Math. Soc., New Ser. 12, 239–246 (1985; Zbl 0572.57002)] and in [J. H. Przytycki and P. Traczyk, Kobe J. Math. 4, No. 2, 115–139 (1987; Zbl 0655.57002)], and is a generalization of both the Alexander polynomial and the Jones polynomial. For \(2\)-bridge knots, a formula to calculate their HOMFLY polynomials was given in [W. B. R. Lickorish and K. C. Millett, Topology 26, 107–141 (1987; Zbl 0608.57009)] and in [T. Kanenobu and T. Sumi, Math. Comput. 60, No. 202, 771–778, S17-S28 (1993; Zbl 0781.57002)]. The formula uses continued fraction representations into a group of \(2\times 2\) matrices.
In this paper, the author starts with a standard and reduced alternating diagram of a \(2\)-bridge knot \(K\) which is obtained from a \(3\)-braid together with an additional straight strand. Let \(T_w(\subset K)\) be the \(2\)-tangle obtained from the \(4\)-braid by capping off two local minima. The main result of this paper states that the “HOMFLY polynomial” of \(T_w\) consists of three polynomials and is calculated by taking a product of \(3\times 3\) matrices. Moreover, the HOMFLY polynomial of \(K\) is easily computed from these three polynomials.


57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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