KNOT POLYNOMIALS FROM Q-DEFORMED ALGEBRAS

We present explicit polynomials of two-knot invariants obtained from q -deformed algebras. Braid-group representations can be obtained from t he R-matrices which in turn arise in q-deformed algebras. A Markov tra ce can be defined for R-matrices based on representations of the g-def ormed algebras su(n)(q) and hence knot polynomials can be defined. In this paper, the properties of coupling coefficients and R-matrices bas ed on each of the {1} and {2} representations for su(n)(q) are used to calculate polynomials for knots of ten or fewer crossings. We develop a new method to calculate the {2}su(n), polynomials. For the {1} repr esentation of su(n)(q), there are five pairs of knots of ten or fewer crossings which have the same polynomial. The exception is where n = 2 . In this case the polynomial is equivalent to the Jones polynomial an d has 14 pairs for knots of ten or fewer crossings. The {2}su(n)(q) po lynomial has four pairs for these knots, each pair is different to the {1}su(n)(q) pairs. Thus, the {2}su(n)(q) polynomial has slightly fewe r pairs than the {1}su(n)(q) polynomial and is significantly better at predicting the amphichirality or non-amphichirality of knots.