Download presentation

Presentation is loading. Please wait.

Published byBrayan Claire Modified over 2 years ago

1
Jonathan Schneider University of Illinois Chicago Virtual 2-knots

2
This is a 2-sphere immersed in R 3. 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 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 virtual

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

10
In a classical 2-knot diagram this is allowedand 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: welded triple point unwelded triple point woven triple point

13
The movie of a triple point is a legal virtual III-move. classicalmixedpure virtual

14
Forbidden triple points have forbidden moves in their movies. unordered classical weldedunweldedwoven

15
2-knot diagrams are equivalent if related by Roseman moves. I-bubble move I-saddle move II-bubble 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. III-move II-bubble moves This works even if some of the crossings are virtual.

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

Similar presentations

Presentation is loading. Please wait....

OK

Secondary Math 3 8-1 Two and Three-Dimensional Objects.

Secondary Math 3 8-1 Two and Three-Dimensional Objects.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on obesity diet management Ppt on content development tools Ppt on energy conservation in india Ppt on ms word 2007 tutorial Respiratory system anatomy and physiology ppt on cells Ppt on sets theory Ppt on telephone exchange in indian railways Ppt on leadership development and succession Ppt on management by objectives Cool backgrounds for ppt on social media