Abstract

We present a systematic construction of traversable wormhole spacetimes featuring two symmetric throats within the framework of General Relativity. Our approach employs the embedding formalism by imposing the analytic relation r - r(0) = K (z(2 )- a)(2) between the radial coordinate and the embedding function, which naturally guarantees the flare-out condition at both throats. This ansatz yields a composite shape function b(r) consisting of two branches: an asymptotically flat branch (epsilon = +1) extending to spatial infinity, and an intermediate tunnel branch (epsilon = -1) with restricted domain r(0) <= r <= r(0) + Ka(2), connecting the two throats positioned at z = +/-root a with separation 2 root a. Analysis of the Einstein field equations reveals that the supporting matter violates the weak, null, and dominant energy conditions throughout the spacetime, while the strong energy condition is identically satisfied with rho + p(r) + 2 p(t )= 0. A critical finding is the direct connection between throat separation and energy density: configurations with 0 < a < 1/(16K r(0)) admit positive energy density at the throats with phantom-type radial pressure (omega(r) < -1) and dark-energy-like effective behavior (omega(eff) < 0), whereas larger separations a > 1/(16 K r(0)) require negative energy density. Notably, configurations with phantom matter at the throats require such small values of a that the intermediate tunnel region becomes almost imperceptible, with the two throats nearly coincident, approaching the limiting case of a single-throat geometry. The geometric structure is visualized through embedding diagrams, which demonstrate that increasing the parameter a produces more pronounced and sharply defined throats. This work establishes that multi-throat wormhole geometries can be systematically generated through embedding techniques, providing a complementary approach to field-theoretic constructions and revealing how topological complexity relates to exotic matter distributions in traversable spacetimes.