3,129 research outputs found
Intersection problem for Droms RAAGs
We solve the subgroup intersection problem (SIP) for any RAAG G of Droms type
(i.e., with defining graph not containing induced squares or paths of length
3): there is an algorithm which, given finite sets of generators for two
subgroups H,K of G, decides whether is finitely generated or not,
and, in the affirmative case, it computes a set of generators for .
Taking advantage of the recursive characterization of Droms groups, the proof
consists in separately showing that the solvability of SIP passes through free
products, and through direct products with free-abelian groups. We note that
most of RAAGs are not Howson, and many (e.g. F_2 x F_2) even have unsolvable
SIP.Comment: 33 pages, 12 figures (revised following the referee's suggestions
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
- …