For Z, this means the trivial path \Gamma=\{\emptyset\}, while for Y\ns_{kl} this means finding the shortest length path from k to l. (If there is no straight line path from k ...For Z, this means the trivial path \Gamma=\{\emptyset\}, while for Y\ns_{kl} this means finding the shortest length path from k to l. (If there is no straight line path from k to l, there will in general be several such minimizing paths.) Note, however, that the presence of the string between sites k and l complicates the analysis of g\ns_\Gamma for the closed loops, since none of the links of \Gamma can intersect the string.
