Friday, September 14, 2012

The blocked-braid groups

I have described in an earlier post my work with Rosebrugh and Sabadini on tangled circuits. A tangled circuit is an arrow on the free braided monoidal category on a monoidal graph where each object (wire) of the monoidal graph is equipped with a commutative frobenius algebra structure.
In that paper we discussed as examples circuits of the form RBS where R and S are two arrows of the monoidal graph, R with domain I and S with codomain I (the unit of the tensor), and where B is a braid, that is a composite of twists tensored with identities. We call such a circuit a blocked braid on n strings.
Notice that in the category of tangled circuits RBS=RB'S does not imply that B=B' as braids. Dirac's belt trick is the fact that if T is a 360 degree rotation of the band of strings then RTTS=R1S(=RS).
A student in Como, Davide Maglia, has been investigating with Sabadini and me the structure of blocked braids as his Laurea Magistrale thesis.
First thing to notice is that blocked braids on n strings form a group, BB_n, the multiplication being (RBS)*(RB'S)=RBB'S, a kind of generalization of the sum of knots.
BB_n is a quotient of the braid group on n strings. The results we have so far are that BB_1=1, BB_2=Z_2, BB_3 has either 12 or 6 elements (we can't tell which), and that BB_n for n>3 is infinite. If BB_3 has twelve elements it is the semidirect product of Z_3 by Z_4 (non-trivial action of Z_4 on Z_3).

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