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We measure the Born cross sections of the process 𝑒+𝑒−→𝐾+𝐾−𝐾+𝐾− at center-of-mass (c.m.) energies, √𝑠, between 2.100 and 3.080 GeV. The data were collected using the BESIII detector at the BEPCII collider. An enhancement at √𝑠=2.232 GeV is observed, very close to the 𝑒+𝑒−→Λ¯Λ production threshold. A similar enhancement at the same c.m. energy is observed in the 𝑒+𝑒−→𝜙𝐾+𝐾− cross section. The energy dependence of the 𝐾+𝐾−𝐾+𝐾− and 𝜙𝐾+𝐾− cross sections differs significantly from that of 𝑒+𝑒−→𝜙𝜋+𝜋−.