Slopes and generalized fillings on torus cusps

Let $\bar{M}$ be a compact connected oriented three-manifold whose boundary is a disjoint union of tori $T_1,\dots ,T_k$. Choose an ordered integral basis $([{\mu }_i],[{\lambda }_i])$ for each $H_1(T_i;\mathbb{Z})$. A slope on $T_i$ is an unoriented isotopy class of essential simple closed curves whose homology class is $p_i[{\mu }_i]+q_i[{\lambda }_i]$ for a primitive pair $(p_i,q_i)\in {\mathbb{Z}}^2$; simultaneous negation gives the same unoriented slope. The primitive-pair correspondence gives exactly one such slope for each primitive pair modulo sign. The ordinary Dehn filling along a tuple of slopes indexed by $J\subseteq \{1,\dots ,k\}$ attaches an oriented solid torus $D^2\times S^1$ to each $T_i$, $i\in J$, by an orientation-reversing boundary homeomorphism that sends $[\mathrm{\partial }{D}^2\times \{\ast \}]$ to the chosen slope. The gluing maps are part of the filling data; $i\mathrm{\notin }J$ means that the cusp is left unfilled, denoted by the parameter $\mathrm{\infty }$.

For a hyperbolic deformation, use signed generalized cone-filling parameters in the one-point compactification ${\mathbb{R}}^2\cup \{\mathrm{\infty }\}$ of the real coefficient plane, with $\mathrm{\infty }$ denoting the complete cusp. The sign convention is fixed by a selected common peripheral eigenline and near-zero eigenvalue-ratio logarithms $u_i,v_i$: at an incomplete cusp the coefficients $g_i=({\alpha }_i,{\beta }_i)$ solve ${\alpha }_i{u}_i+{\beta }_i{v}_i=2\pi i$. Reversing the selected eigenline negates both logarithms and the coefficient vector. Forgetting this selection gives the intrinsic generalized invariant in $({\mathbb{R}}^2\mathrm{/}\{g\sim -g\})\cup \{\mathrm{\infty }\}$; the unquotiented sphere is its signed parameter space. A primitive integral pair $(p_i,q_i)$ is an ordinary smooth filling coefficient. If a realized generalized parameter lies on the ray $g_i=r_i(p_i,q_i)$ with $r_i>0$ and $(p_i,q_i)$ primitive integral, its topological core direction is the ordinary slope $(p_i,q_i)$ and its cone angle is $2\pi \mathrm{/}{r}_i$. Only $r_i=1$ is the ordinary smooth filling; $r_i>1$ describes a cone completion (an orbifold completion when $r_i$ is an integer) and does not turn the nonprimitive vector $g_i$ into a slope.

Primitive-pair correspondence on a torus

William P. Thurston, The Geometry and Topology of Three-Manifolds, Chapter 4, §4.5

William P. Thurston, The Geometry and Topology of Three-Manifolds, Chapter 5, §5.8

Danny Calegari, Kleinian Groups, §2.4.1