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--- abstract: | Let $\GG = \langle \AA,\BB\rangle$ be a non-elementary two generator subgroup of the isometry group of $\HH^2$. If $G$ is discrete, free and generated by two hyperbolic elements with disjoint axes, its quotient is a pair of pants and in [@GKwords] we produced a recursive formula for the number of essential self intersections (ESIs) of any primitive geodesic on the quotient. An alternative non-recursive proof of the formula was given in [@Vidur]. We used the formula to give a geometric description of how the primitive geodesic winds around the “cuffs" of the pants. On a pair of pants, an ESI is a point of a “seam" where a primitive geodesic has a self-intersection. Here we generalize the definition of ESI’s to non-elementary purely loxodromic two generator groups $G \subset \Isom(\HH^3)$ which are discrete and free. We also associate an open geodesic to each generalized ESI we call a [*Connector.*]{} In the quotient manifold the connectors correspond to “opened up self-intersections". We then define a class of groups we call [*Winding groups*]{}; the connectors for these groups give us a way generalize the concept of winding of primitive geodesics in the quotient manifold to this context. For winding groups, the ESI’s and connectors satisfy the same formula as they do in the two-dimensional case. Our techniques involve using the [*stopping generators*]{} we defined in [@GKwords] for the group $\GG$ to make it a model. We then use the model to obtain an algebraic rather than geometric definition for ESI’s and generalize from the algebraic definition. address: - 'Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07079' - 'Department of Mathematics, The Graduate Center and Lehman College, CUNY, New York, NY 10016' author: - Jane Gilman - Linda Keen title: 'Winding and Unwinding and essential intersections in $\HH^3$' --- Introduction ============ In [@GKwords] we considered some geometric implications of the Gilman-Maskit discreteness algorithm, [@GM], for non-elementary two generator subgroups of $\Isom(\HH^2)$ whose generators were hyperbolic with disjoint axes and whose product was also hyperbolic. We showed that when the algorithm determines that the group is discrete and free, the quotient is a three holed sphere and the algorithm has determined discreteness by finding a special pair of [*stopping generators*]{} that correspond to the only three simple geodesics on the quotient surface, those around the waist and the cuffs. Every other geodesic on the quotient winds around the waist and cuffs and has self-intersections. We concentrate our attention on the [*primitive geodesics*]{} on the quotient. A primitive element of the fundamental group of the surface is one that together with one other element generates the fundamental group. A [*primitive curve*]{} is the geodesic in the free homotopy class of such a primitive element. A pair of such geodesics coming from a pair of elements that generate the fundamental group is a [*primtive pair*]{}. These conjugacy classes of primitive geodesics can be identified with the set of rational numbers, [@Enumeration; @KS1]. In discreteness considerations, it is standard to consider