On a parameterized family of twist quantum coalgebras and the bracket polynomial
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 a submitted paper of L. H. Kauffman and D. E. Radford. In this paper, the author uses the simple coalgebra \(C_n(k)\) (i.e. the dual of the algebra \(M_n(k)\) of \(n\times n\) matrices over \(k\)) as a starting point for constructing twist quantum coalgebras. Among the twist quantum coalgebras, a parameterized family is constructed related to the Jones polynomial. This paper is organized as follows. In Section 1, the author sets forth the rather minimal essential coalgebra prerequisites needed in the sequel. (Twist) quantum coalgebras are defined, the produced invariants are described in detail, and examples are given. This section also includes an explicit description of the 4-dimensional twist quantum coalgebra, which produces the bracket polynomial for knots. Section 2 is devoted to describing the construction of \(C_T\) from the triples \((C,b,T)\), found in the author's other paper which will appear in J. Algebra, where \(C\) is a coalgebra over \(k\) with automorphism \(T\) and \(T\)-form \(b\), and the relationship between \(C_T\) and \(C\) and their invariants when \((C,b,T)\) arises from a quantum coalgebra structure on \(C\). In Section 3, such triples \((C,b,T)\) are studied where \(C=C_n(k)\) for some \(n\geq 1\) and \(T\) is a diagonalizable coalgebra automorphism of \(T\) of a certain type. Results of this section provide the basis for the construction respectively in Section 4 of the family of twist quantum coalgebras \(C(a,b,c)\) and in Section 5 of parameterized families of twist quantum coalgebras of dimension \(2n^2\) for \(n>2\). In Section 6, the author discusses the Laurent polynomial invariant arising from the parameterized family of twist quantum coalgebras \(C(a,b,c)\).
scientific article; zbMATH DE number 1423605