Geek Challenge: Malleable Mystery

January’s Geek challenge is from the field of Topology, which is the study of continuity and connectivity of shapes. This field of math gives us such gems as the Hairy Ball Theorem.  

A Continuous Deformation of a shape is what you can do to a piece of clay if you never sever the clay, or rejoin the clay at any point. You can squish, bend and twist all you want at any point. Using Continuous Deformation, a coffee cup can be transformed into a doughnut, as demonstrated in this animation:

Which of the following three scenarios are possible using Continuous Conversion?

1).
Connected dual-ring part A can be converted in to disconnected dual-ring part B around the perimeter B.

2).  
Dual-ring part A double hung on a ring can be converted to dual-ring part B single hung around the ring. In this one, the ring can’t deform – only the dual-ring part.

3).  
Pretzel-like part A into 8-shaped part B.

Which scenarios are possible?

A: Only #2.

B: #1 and #3, but not #2.

C:  None of them

D:  All three of them