Surfaces in 3-space that do not lift to embeddings in 4-space

Abstract

Abstract.A necessary and sufficient condition for an immersed surface in 3-space to be lifted to an embedding in 4-space is given in terms of colorings of the preimage of the double point set.Giller's example and two new examples of non-liftable generic surfaces in 3-space are presented. One of these examples has branch points. The other is based on a construction similar to the construction of Giller's example in which the orientation double cover of a surface with odd Euler characteristic is immersed in general position. A similar example is shown to be liftable with an explicit lifting given.The problem of lifting is discussed in relation to the theory of surface braids. Finally, the orientations of the double point sets are studied in relation to the lifting problem.

Computations of Quandle Cocycle Invariants of Knotted Curves and Surfaces

“…Stills (1,1), (1,2), (6,7), and (6,8) represent critical levels in which simple closed curves are born or die. In particular, the still (1,1) consists of a single dot where a birth occurs.…”

“…These curves are colored a and b with the curve colored b nested inside the curve colored a. Other critical points (saddle points) occur between stills (1,7) and (1,8) and between (6,1) and (6,2). Since these are all the critical points the resulting surface depicted is a sphere.…”

Crossing distances of surface-knots

“…It is known that every immersed circle in R 2 can be lifted into R 3 as a knot. It is proved that there are some immersed surfaces in R 3 , which cannot be lifted into R 4 [4,3,11]. It is not known whether or not every liftable generic surface is realised as a trivial surface in R 4 .…”

On Crossing Changes for Surface-Knots

“…Let S a = ∪ n i=1 cl(s i a ) and let S b = ∪ n i=1 cl(s i b ). The union S a ∪ S b is called the double decker set (see [1] and [2] for details).…”