virtual 2-knots
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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Jonathan Schneider
University of Illinois Chicago
Virtual 2-knots
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This is a 2-sphere immersed in R3. Its cross section is a closed curve.
![Page 3: Virtual 2-knots](https://reader035.vdocuments.us/reader035/viewer/2022062805/56814d0c550346895dba46e3/html5/thumbnails/3.jpg)
I cut a slice out so you can see inside.
![Page 4: Virtual 2-knots](https://reader035.vdocuments.us/reader035/viewer/2022062805/56814d0c550346895dba46e3/html5/thumbnails/4.jpg)
This is a 2-knot diagram. Its cross section is a classical 1-knot diagram.
![Page 5: Virtual 2-knots](https://reader035.vdocuments.us/reader035/viewer/2022062805/56814d0c550346895dba46e3/html5/thumbnails/5.jpg)
This is a virtual 2-knot diagram.Its cross-section is a
virtual 1-knot diagram.
![Page 6: Virtual 2-knots](https://reader035.vdocuments.us/reader035/viewer/2022062805/56814d0c550346895dba46e3/html5/thumbnails/6.jpg)
The crossings in a virtual 2-knot diagram are either
classical or virtual.
Classical crossing 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 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 allowed and 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.
classical mixed pure virtual
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Forbidden triple points have forbidden moves in their movies.
unordered classical welded unwelded woven
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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.