In the paper Blocked-braid Groups, submitted to Applied Categorical
Structures, the present authors together with Davide Maglia introduced the
blocked-braid groups BB_n on n strands, and proved that a blocked torsion has
order either 2 or 4. We conjectured that the order was actually 4 but our
methods in that paper, which involved introducing for any group G a braided
monoidal category of tangled relations, were inadequate to demonstrate this
fact. Subsequently Davide Maglia in unpublished work investigated exactly what
part of the structure and properties of a group G are needed to permit the
construction of a braided monoidal category with a tangle algebra and was able
to distinguish blocked two-torsions from the identity.
In this paper we present a simplification of his answer, which turns out to
be related to the notion of rack. We show that if G is a rack then there is a
braided monoidal category TRel_G generalizing that of the above paper. Further
we introduce a variation of the notion of rack which we call irack which yields
a tangle algebra in TRel_G. Iracks are in particular racks but have in addition
to the operations abstracting group conjugation also a unary operation
abstracting group inverse. Using iracks we obtain new invariants for tangles
and blocked braids permitting us to present a proof of Maglia's result that a
blocked double torsion is not the identity.
This work was presented at the Conference in Memory of Aurelio Carboni,
Milan, 24-26 June 2013