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**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.

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.

Reviewer: Toshio Saito (Joetsu)

### MSC:

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

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\textit{B.-H. Kwon}, Rocky Mt. J. Math. 46, No. 1, 243--260 (2016; Zbl 1347.57015)

### References:

[1] | C.C. Adams, The knot book , W.H. Freeman and Co., New York, 1994. · Zbl 0840.57001 |

[2] | J. Birman, Braids, links and mapping class groups , Ann. Math. Stud. 82 , Princeton University Press, 1974. |

[3] | P. Freyd, D. Yetter, W.B.R. Lickorish, K. Millett and A. Ocneanu, A new polynomial invariant of knots and links , Bull. Amer. Math. Soc. 12 (1985), 239-246. · Zbl 0572.57002 |

[4] | J.R. Goldman and L.H. Kauffman, Rational tangles , Adv. Appl. Math. 18 (1997), 300-332. · Zbl 0871.57002 |

[5] | T. Kanenobu and T. Sumi, Polynomial invariants of \(2\)-bridge knots through \(22\) crossings , Math. Comp. 60 (1993), 771-778, S17-S28. · Zbl 0781.57002 |

[6] | L.H. Kauffman and S. Lambropoulou, On the classification of rational tangles , Adv. Appl. Math. 33 (2004), 199-237. · Zbl 1057.57006 |

[7] | W.B.R. Lickorish and K.C. Millett, A polynomial invariant of oriented links , Topology 26 (1987), 107-141. · Zbl 0608.57009 |

[8] | P. Przytycki and P. Traczyk, Invariants of links of Conway type , Kobe J. Math. 4 (1988), 115-139. · Zbl 0655.57002 |

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