Dissecting the 2-sphere by immersions

Abstract

Abstract. The self intersection of an immersion i : S 2 → R 3 dissects S 2 into pieces which are planar surfaces (unless i is an embedding). In this work we determine what collections of planar surfaces may be obtained in this way. In particular, for every n we construct an immersion i : S 2 → R 3 with 2n triple points, for which all pieces are discs.

A characterisation of alternating knot exteriors

“…Exactly one component of S 0 \ B 0 contains the connected graph f −1 0 (Γ). Using an Euler characteristic argument, Nowik [18] points out that…”

Dissecting the torus by immersions

“…Nowik proved the following theorem in [2]. Our Theorem 2 is a torus version of Nowik's theorem above.…”

“…Therefore, n and {b k } ∞ k=1 satisfy (P). Then, by Nowik's Theorem 3 (see also [2]), the sequence n, {b k } ∞ k=1 is realized by a generic immersion j : N (M( j)) that is homeomorphic to S 2 ( ) . By attaching a handle to K as in Fig.…”

How do curved spheres intersect in 3-space?

“…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]…”