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Virtual 2-knots

Virtual 2-knots. Jonathan Schneider. University of Illinois Chicago. This is a 2-sphere immersed in R 3. 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.

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Virtual 2-knots

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  1. Virtual 2-knots Jonathan Schneider University of Illinois Chicago

  2. This is a 2-sphere immersed in R3. Its cross section is a closed curve.

  3. I cut a slice out so you can see inside.

  4. This is a 2-knot diagram. Its cross section is a classical 1-knot diagram.

  5. This is a virtual 2-knot diagram. Its cross-section is a virtual 1-knot diagram.

  6. The crossings in a virtual 2-knot diagram are either classical or virtual. Classical crossing Classical crossing Virtual crossing

  7. Crossings are closed curves or open intervals. These disks cross in a circle. This tube crosses itself in an interval.

  8. Interval crossings end at pinch points. classical classical virtual

  9. Some crossings contain triple points. Example: Three intersecting spheres

  10. In a classical 2-knot diagram this is allowed and this is forbidden.

  11. In a virtual 2-knot diagram, these are also allowed: pure virtual triple point mixed triple point

  12. …but these are forbidden: unwelded triple point welded triple point woven triple point

  13. The movie of a triple point is a legal virtual III-move. classical mixed pure virtual

  14. Forbidden triple points have forbidden moves in their movies. unordered classical welded unwelded woven

  15. 2-knot diagrams are equivalent if related by Roseman moves. I-bubble move II-bubble move I-saddle move II-saddle move

  16. These Roseman moves involve triple points. III move branch pass quadruple point

  17. Use the Roseman moves to separate the three spheres. This works even if some of the crossings are virtual. III-move II-bubble moves

  18. Thank you. Jonathan Schneider University of Illinois Chicago jschne9@uic.edu Another virtual 2-knot.

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