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In this article we study from a non-perturbative point of view theentanglement of two directed polymers subjected to repulsive interactions givenby a Dirac $\delta-$function potential. An exact formula of the so-calledsecond moment of the winding angle is derived. This result is used to provide athorough analysis of entanglement phenomena in the classical system of twopolymers subjected to repulsive interactions and related problems. Noapproximation is made in treating the constraint on the winding angle and therepulsive forces. In particular, we investigate how repulsive forces influencethe entanglement degree of the two-polymer system. In the limit of idealpolymers, in which the interactions are switched off, we show that our resultsare in agreement with those of previous works.Comment: 34 pages, LaTeX + RevTeX 4, minor typos corrected, 2 figures added, a  major revision of Section

Directed polymers with constrained winding angle