Calculation of Knot Polynomials for Unknotted Knots

Abstract

Recently, we have succeeded in constructing algebraic equations among knot and link polynomials. In this paper we report that the polynomials for unknot are really solved exactly from the equations, in the case of SU(N) and E 6• Such data are expected to give us a new insight of the Chern· Simons theory because knot polynomials can be regarded as vacuum expectation values of Wilson loop operators in the Chern· Simons theory. Knot polynomials 1H ) can be regarded as vacuum expectation values of Wilson loop operators in 2+ 1 dimensional Chern-Simons theories. 5H9 ) Interestingly enough, we could evaluate them exactly by using the relationship5),20)-22) between the Chern­ Simons theories and 1 + 1 dimensional rational conformal field theories. 23 )-29) Recently, we have shown that vacuum expectation values of Wilson loop opera­ tors in the Chern-Simons theory satisfy algebraic equations if the gauge group is compact and simple. 30 ) In this paper we report briefly typical examples. Algebraic equations for vacuum expectation values of unknotted Wilson loop operators are derived and solved exactly, in the case of SU(N) and E 6 • Because this paper is devoted to rapid communications, we have omitted explanations of many interesting topics. They are discussed in detail in Ref. 30). Our calculations show that vacuum expectation value of unknotted Wilson loop operator in p representation of gauge group G, which is denoted by Eo(p), is nothing but a q-dimensions for Uq(G).**) Consequently, it has close relations between char­ acter formulas of affine Lie algebra, S-matrix of modular transformation, Verlinde algebra, and so on. Although no argument about these properties of Eo(p) is included in this paper, they are discussed in Ref. 30). Here we stress one point. Explicit evaluation of Eo(p) must be useful to eluci­ date some features of the Chern-Simons theory. For example, our computation here is regarded as the manifestations of quantum group symmetry which is hidd~n in the Chern-Simons theory, although it is rather indirect. We think, however, such an observation is important because it is almost impossible to foresee the existence of the quantum group symmetry at the level of Chern-Simons action. As is well known, 2+ 1 dimensional gravity theories are also formulated as the Chern-Simons theories, though, in these cases, the gauge group must be non-compact. Our method might be applicable also for such cases. It means that our computations