= Genus-one gluing model of a lens space
{title2=$\mu_2\mapsto-q\mu_1+p\lambda_1$}
The quotient model of a <lens space> splits into two <solid tori>. In the boundary angle lattice $\mathbb Z^2+\mathbb Z(1/p,q/p)$, choose $\mu_1=(0,1)$ and $\lambda_1=(1/p,q/p)$. The other meridian is $(1,0)=-q\mu_1+p\lambda_1$. A second column completing this primitive slope to a determinant-one matrix specifies a gluing map after reversing one boundary-basis convention. Different completions differ by a meridional <Dehn twist> extending across the second solid torus, so they give the same <three-manifold>.
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