An Infinite Suite of Links–Gould Invariants

This paper describes a method to obtain state model parameters for an infinite series of Links–Gould link invariants LG, based on quantum R matrices associated with the (0̇m | α̇n) representations of the quantum superalgebras Uq [gl(m|n)]. Explicit details of the state models for the cases n = 1 and m = 1, 2, 3, 4 are supplied. Some evaluations of the new link invariants are provided, as are some of their gross properties. 1 Overview In 1992, Jon Links and Mark Gould [13] described a method for constructing link invariants from quantum superalgebras. That work stopped short of evaluations of the invariants due to want of an efficient computational method. In 1999, the author, in collaboration with Jon Links and Louis Kauffman [6], first evaluated a twovariable example of one these invariants, using a state model. We used the (0, 0 |α) representations of Uq[gl(2|1)], and labeled our resulting (1, 1)-tangle invariant LG, ‘the Links–Gould invariant’. In that paper, and subsequently in [4], we showed that whilst LG would detect neither inversion nor mutation, it was still able to distinguish all prime knots of up to 10 crossings, making it more powerful than the HOMFLY and Kauffman invariants. Here, we generalise the notation, denoting LG as “the Links–Gould invariant associated with the (0̇m | α̇n) representation of Uq[gl(m|n)]”. For the case n = 1, we will write LG ≡ LG, so our previous invariant LG was in fact LG. This generalisation is motivated by the automation of a procedure to construct the appropriate R matrices [3, 5]; previously, we were limited to the m = 2 case, for which the R matrix had been calculated by hand. Further, we explicitly demonstrate the construction of state model parameters for LG, illustrating our results for LG, for the cases m = 1, 2, 3, 4. Further, we describe some of the gross properties of these invariants, and provide a limited set of evaluations of them. Although these invariants LG are not more powerful in their gross properties than LG (they can detect neither inversion nor mutation), they are expected to distinguish many more knots as the degree of the polynomials that they yield increases with m and n. Perhaps more significantly, the development of the current formalism is pointing the way to automation of the evaluation of more general classes of quantum link invariants, and a discussion of this is provided. RIMS, Kyoto University 606-8502, Japan. [email protected] 1 2 Quantum Superalgebra State Models Corresponding to each finite dimensional highest weight representation of each quantum superalgebra, there exists a quantum link invariant [2, 6, 13]. Here, we describe the construction of parameters for state models and their use in the evaluation of link invariants for Uq[gl(m|n)] using representations π ≡ πΛ of highest weight Λ. This material is of course applicable to ordinary quantum algebras. (In §4, we specialise this material to the case Λ = (0̇m | α̇n), and in §5, setting n = 1, we demonstrate explicit results for the examples m = 1, 2, 3, 4.) We may construct a state model for evaluation of these invariants from explicit knowledge of two parameters: • the (tensor product) representation Ř ≡ (π ⊗ π)Ř of the quantum R matrix Ř, and • the representation of the Cartan element S ≡ π(qρ). As Ř necessarily satisfies the QYBE, σ , κσŘ (for any scalar constant κσ) realises a representation of the braid generator. This follows as abstract tensors built from σ are invariant under the second and third Reidemeister moves, hence we may construct representations of arbitrary braids from σ. Our state model also requires us to represent left handles C, i.e. arcs closing braids to form links. As all links may be represented by braids combined with left handles, together these are sufficient parameters. To ensure that our resulting invariants are invariants of ambient isotopy, we must select C to ensure that the resulting abstract tensors built from σ and C are invariant under the first Reidemeister move. To this end, we apply (a grading-stripped version of) the following result [15, Lemma 2] (see also [13]): (I ⊗ str)[(I ⊗ qρ)σ] = KI, where str is the supertrace, and K is some constant depending on the normalisation of σ. Thus, for any scalar constant κC , setting C , κCS allows us to represent left handles. Below, we demonstrate how to select κσ and κC such that the abstract tensor associated with removal of an isolated loop is invariant under the first Reidemeister move. Figure 1 shows that for σ and C to satisfy the first Reidemeister move, they must satisfy (Einstein summation convention): C c · σ ca db = δ a b = C d c · σ ca db, (1) where the definitions of κσ and κC yield: σ = κ −1 σ Ř , and C = κ C S . For quantum superalgebras, the resulting R matrices are in fact graded, and satisfy a graded QYBE. It is a simple matter to strip out this grading [5], yielding Ř which satisfies the usual, ungraded QYBE. Here, we implicitly use grading-stripped versions of Ř and S. The Ř supplied in [5] are normalised such that limq→1 Ř is a (graded) permutation matrix. Scaling by κσ does not change that. We shall occasionally use the notation x to mean x, in particular we shall write q ≡ q, σ ≡ σ and C ≡ C (the right handle). Doing this allows us to omit superfluous “+” signs, e.g. we shall write C ≡ C for the left handle. 2