On the HOMFLY polynomial of 4-plat presentations of knots

Abstract

Abstract. In this paper, I give a method to calculate the HOMFLY polynomials of twobridge knots by using a representation of the braid group B 4 into a group of 3×3 matrices.Also, I will give examples of a 2-bridge knot and a 3-bridge knot that have the same Jonespolynomial, but different HOMFLY polynomials. 1. IntroductionIn 1985, Hoste-Ocneanu-Millett-Freyd-Lickorish-Yetter [3] had discovered the HOMFLYpolynomial which is a 2-variable oriented link polynomial P L (a,m) motivated by the Jonespolynomial. Also, Prztycki and Traczyk [7] independently discovered the HOMFLY polyno-mial. The calculation of the HOMFLY polynomial is based on the HOMFLY skein relationsas follows.(1) P(L) is an isotopy invariant.(2) P(unknot)=1.(3) a·P(L + ) +a −1 ·P(L − )+m·P(L 0 ) = 0.L L _ L+ 0 Figure 1.Actually, Lickorish and Millett [6] gave a formula to calculate the HOMFLY polynomials ofrational knots by using a representation of the continued fraction of a rational knot into agroup of 2× 2 matrices. (See Proposition 14.)In this paper, I use the Kauffman blacket polynomial and the plat presentation of knots tocalculate the HOMFLY polonomials of rational knots by using a representation of the braidgroup B