On a parameterized family of twist quantum coalgebras and the bracket polynomial

Abstract

Quantum coalgebras over a field k are structures that produce regular isotopy invariants of 1-1 tangles, and twist quantum coalgebras over k are structures that produce regular isotopy invariants of knots. These structures are introduced and studied in [9]. In [6] and [8] certain families of quantum coalgebras and their resulting invariants of 1-1 tangles are examined in detail. In this paper we use the simple coalgebra C,(k), the dual of the algebra Mn(k) of n x n matrices over k, as a starting point for constructing twist quantum coalgebras. Among the twist quantum coalgebras we construct is a parameterized family related to the Jones polynomial in an interesting way.The Jones polynomial can be computed by the (normalized) bracket polynomial [5, Theorem 5.2]. The bracket polynomial (for knots) in turn can be calculated from a quantum algebra structure on M2(C), which is described in [5, p. 580]. The dual of this quantum algebra is a certain twist quantum coalgebra structure on C2(C), which is discussed in [9, Sect. 8]. Thus the Jones polynomial for knots can be computed from a twist quantum coalgebra structure on C2(C).

Oriented Quantum Algebras and Invariants of Knots and Links

“…A rather tedious calculation shows that the set of equations ( 10)-( 13) is equivalent to (qa.3) under the assumption that ( 6) and ( 7) hold. By passing to the dual coalgebra C n (k) = M n (k) * , one sees that this equivalence is established in [18,Lemma 4]. Using Proposition 6 and retracing the proof of [18,Theorem 2] we can conclude that the set of statements ( 7)-( 9) is equivalent to ρ is invertible and (qa.1) holds for ρ and t under the assumption that (6) holds.…”

“…By passing to the dual coalgebra C n (k) = M n (k) * , one sees that this equivalence is established in [18,Lemma 4]. Using Proposition 6 and retracing the proof of [18,Theorem 2] we can conclude that the set of statements ( 7)-( 9) is equivalent to ρ is invertible and (qa.1) holds for ρ and t under the assumption that (6) holds. For the sake of completeness we will sketch a proof that ( 7)-( 9) are collectively equivalent to ρ is invertible, (qa.1) holds for ρ and t under the assumption that (6) holds.…”

“…Then E ı 1≤ı ≤n is the standard basis for M n k and E ı E m = δ  E ım for all 1 ≤ ı  m ≤ n. The notation for the scalar b c is meant to suggest a product. That ρ a B C satisfies the quantum Yang-Baxter equation follows by [18,Lemma 4 and (37)]. …”

A separation result for quantum coalgebras with an application to pointed quantum coalgebras of small dimension

“…Under the assumption that (7) and (8) The condition/3 is invertible and/3-1(x, y) =/3(S(x), y) is equivalent to b is invertible and the equations (10) for all c, d ~ C which, since b is invertible, account for the four cases which arise from the equation…”

“…In [10] we use our construction to study a class of quantum coalgebras which includes the quantum coalgebra which gives rise to the Jones polynomial. These coalgebras produce polynomial invariants of 1-1 tangles and knots.…”