Quantum automata, braid group and link polynomials

The spin--network quantum simulator model, which essentially encodes the (quantum deformed) SU(2) Racah--Wigner tensor algebra, is particularly suitable to address problems arising in low dimensional topology and group theory. In this combinatorial framework we implement families of finite--states and discrete--time quantum automata capable of accepting the language generated by the braid group, and whose transition amplitudes are colored Jones polynomials. The automaton calculation of the polynomial of (the plat closure of) a link L on 2N strands at any fixed root of unity is shown to be bounded from above by a linear function of the number of crossings of the link, on the one hand, and polynomially bounded in terms of the braid index 2N, on the other. The growth rate of the time complexity function in terms of the integer k appearing in the root of unity q can be estimated to be (polynomially) bounded by resorting to the field theoretical background given by the Chern-Simons theory.Comment: Latex, 36 pages, 11 figure

On the diameter of the rotation graph of binary coupling trees

AbstractA binary coupling tree on n+1 leaves is a binary tree in which the leaves have distinct labels. The rotation graph Gn is defined as the graph of all binary coupling trees on n+1 leaves, with edges connecting trees that can be transformed into each other by a single rotation. In this paper, we study distance properties of the graph Gn. Exact results for the diameter of Gn for values up to n=10 are obtained. For larger values of n, we prove upper and lower bounds for the diameter, which yield the result that the diameter of Gn grows like nlg(n)

On the Diameter of the Rotation Graph of Binary Coupling Trees

A binary coupling tree on n+1 leaves is a 0--2-tree in which each leaf has a distinct label. The rotation graph Gn is defined as the graph of all binary coupling trees on n + 1 leaves, with edges connecting trees that can be transformed into each other by a single rotation. In this paper we study distance properties of the graph Gn . Exact results for the diameter d(Gn ) for values upto n = 10 are obtained. For larger values of n we prove upper and lower bounds for the diameter, which yield the result that the diameter d(Gn ) grows like n lg(n). Corresponding author : V. Fack, Department of Applied Mathematics and Computer Science, University of Ghent, Krijgslaan 281-S9, B-9000 Gent, Belgium. Tel. ++ 32 9 2644808; Fax ++ 32 9 2644995; E-mail [email protected]. 1 Research Associate of the Fund for Scientific Research -- Flanders (Belgium) 1 Introduction Binary coupling trees arise in the context of the quantum theory of angular momentum, where they represent coupling schemes in..