Contact structures: dz+xdy-ydx

Some views of the planes in the kernel of the 1-form dz+xdy-ydx

Cosmetic surgery on the Figure Eight Sister

Cosmetic surgery on the Figure Eight Sister, originally uploaded by epsilon_is_afraid_of_zeta.

The double branched cover of the closure of the braid on the left is the lens space L(5,1). The braid axis lifts to the Figure Eight Sister. Adding in two full twists downstairs is equivalent to doing +1 surgery on the knot up stairs. After braid moves, we obtain the mirror.

Hence the figure eight sister exterior has two fillings yielding homeomorphic lens spaces.

Scharlemann cycles

Connect sum

Blackboard pictures

Grid diagram and trefoil

Twisted Saddles

Figure Eight and Braid

Hmmm.... I'll have to figure out how to put these in the correct order.

Contact structure

ctctstr4, originally uploaded by epsilon_is_afraid_of_zeta.

Here we see one view of some 2-planes of the standard contact structure.