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A 1-factorization (or parallelism) of the complete graph with loops (P, E , ) is called polar if each 1-factor (parallel class) contains exactly one loop and for any three distinct vertices x1, x2, x3, if {x1} and {x2, x3} belong to a 1-factor then the same holds for any permutation of the set {1, 2, 3}. To a polar graph (P, E , ) there corresponds a polar involution set (P, I), an idempotent totally symmetric quasigroup (P, * ), a commutative, weak inverse property loop (P, +) of exponent 3 and a Steiner triple system (P, B).We have: (P, E , ) satisfies the trapezium axiom ⇔ ∀α ∈ I : αIα = I ⇔ (P, * ) is self-distributive ⇔ (P, +) is a Moufang loop ⇔ (P, B) is an affine triple system; and: (P, E , ) satisfies the quadrangle axiom ⇔ I 3 = I ⇔ (P, +) is a group ⇔ (P, B) is an affine space. Mathematics Subject Classification (2000). 20N05, 05C70, 51E10.

Starting from an involutorial difference loop (P,+) of order n we construct a new loop (L,.) in the same class and possessing 2n elements. The construction was induced by studying a correspondence connecting such loops with complete graphs with parallelism and regular involution sets. We discuss conditions on (P,+) ensuring that the loop (L,.) is a K-loop and we give explicit examples of K-loops obtained with this method. Further generalizations of this technique are proposed as well