How do curved spheres intersect in 3-space?

Abstract

The following problem was proposed in 2010 by S. Lando. Let M and N be two unions of the same number of disjoint circles in a sphere. Do there always exist two spheres in 3-space such that their intersection is transversal and is a union of disjoint circles that is situated as M in one sphere and as N in the other? Union M of disjoint circles is situated in one sphere as union M of disjoint circles in the other sphere if there is a homeomorphism between these two spheres which maps M to M .We prove (by giving an explicit example) that the answer to this problem is "no". We also prove a necessary and sufficient condition on M and N for existing of such intersecting spheres. This result can be restated in terms of graphs. Such restatement allows for a trivial brute-force algorithm checking the condition for any given M and N .It is an open question if a faster algorithm exist. The Lando ProblemWe work entirely in the piecewise-linear (PL) category 1 .Suppose M and M are the unions of the same number of disjoint circles in spheres S and S . Then M is situated in S as M in S if there is a homeomorphism f : S → S such that f (M ) = M .The following problem suggested by S. Lando was one of the (unsolved) problems at the Moscow State University mathematical tournament for students and young professors 2010 ([1], problem MB-8).Let M and N be two unions of the same number of disjoint circles in a sphere. Do there exist two spheres in 3-space whose intersection is transversal and is a union of disjoint circles that is situated as M in one sphere and as N in the other?This problem appeared in the discussion of related papers [3], [4], [5].In this paper we prove that the answer to Lando problem is "no" by giving an explicit example.1 arXiv:1210.7361v2 [math.GT]