Isotopy of Loops - The Geometric Interpretation of Isotopy

The Geometric Interpretation of Isotopy

Given a loop L, one can define an incidence geometric structure called a 3-net. Conversely, after fixing an origin and an order of the line classes, a 3-net gives rise to a loop. Choosing a different origin or exchanging the line classes may result in nonisomorphic coordinate loops. However, the coordinate loops are always isotopic. In other words, two loops are isotopic if and only if they are equivalent from geometric point of view.

The dictionary between algebraic and geometric concepts is as follows

• The group of autotopism of the loop corresponds to the group direction preserving collineations of the 3-net.
• Pseudo-automorphisms correspond to collineations fixing the two axis of the coordinate system.
• The set of companion elements is the orbit of the stabilizer of the axis in the collineation group.
• The loop is G-loop if and only if the collineation group acts transitively on the set of point of the 3-net.
• The property P is universal if and only if it is independent on the choice of the origin.