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 Chicagojschne9@uic.edu