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Jonathan Schneider University of Illinois Chicago Virtual 2-knots

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This is a 2-sphere immersed in R 3. Its cross section is a closed curve.

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I cut a slice out so you can see inside.

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

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

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

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

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Interval crossings end at pinch points. classical virtual

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Some crossings contain triple points. Example: Three intersecting spheres

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In a classical 2-knot diagram this is allowedand this is forbidden.

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

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

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The movie of a triple point is a legal virtual III-move. classicalmixedpure virtual

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Forbidden triple points have forbidden moves in their movies. unordered classical weldedunweldedwoven

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

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

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

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Thank you. Another virtual 2-knot. Jonathan Schneider University of Illinois Chicago

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