Jonathan Schneider
University of Illinois Chicago
Virtual 2-knots
This is a 2-sphere immersed in R3. Its cross section is a closed curve.
I cut a slice out so you can see inside.
This is a 2-knot diagram. Its cross section is a classical 1-knot diagram.
This is a virtual 2-knot diagram.Its cross-section is a
virtual 1-knot diagram.
The crossings in a virtual 2-knot diagram are either
classical or virtual.
Classical crossing Classical crossing Virtual crossing
Crossings are closed curves or open intervals.
These disks cross in a circle. This tube crosses itself in an interval.
Interval crossings end at pinch points.
classical classical virtual
Some crossings contain triple points.
Example: Three intersecting spheres
In a classical 2-knot diagram
this is allowed and this is forbidden.
In a virtual 2-knot diagram, these are also allowed:
pure virtual triple point mixed triple point
…but these are forbidden:
welded triple point
unwelded triple point
woven triple point
The movie of a triple point is a legal virtual III-move.
classical mixed pure virtual
Forbidden triple points have forbidden moves in their movies.
unordered classical welded unwelded woven
2-knot diagrams are equivalent if related by Roseman moves.
I-bubble move
I-saddle move
II-bubble move
II-saddle move
These Roseman moves involve triple points.
III move
branch pass
quadruple point
Use the Roseman moves to separate the three spheres.
III-move
II-bubble moves
This works even if some of the crossings are virtual.
Thank you.
Another virtual 2-knot.
Jonathan SchneiderUniversity of Illinois [email protected]