Hexagonal circular 3-webs with polar curves of degree three

Lie sphere geometry describes circles on the unit sphere by polar points of these circles. Therefore a one parameter family of circles corresponds to a curve and a 3-web of circles, i.e., 3 foliations by circles, is fixed by 3 curves. We call the union of these curves the polar curve and show how analysis of the singular set of hexagonal 3-webs helps to classify circular hexagonal 3-webs with polar curves of degree 3. Many of the found webs are new. The presented results mark the progress in the Blaschke-Bol problem posed almost one hundred years ago. More detail in arXiv:2306.11707.