We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you!
Presentation is loading. Please wait.
Published byBrayan Claire
Modified over 2 years ago
Jonathan Schneider University of Illinois Chicago Virtual 2-knots
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.
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 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 virtual
Some crossings contain triple points. Example: Three intersecting spheres
In a classical 2-knot diagram this is allowedand 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. classicalmixedpure virtual
Forbidden triple points have forbidden moves in their movies. unordered classical weldedunweldedwoven
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 Schneider University of Illinois Chicago email@example.com
There are some mirrors that distort the reflected image. Cosmetic mirrors magnify things, and other mirrors make things look smaller.
Geometric Solids EQ: What are the most common types of solids, what are cross sections and solids of revolution?
Volumes 7.3. Finding Volume Using the Cross Section Think of a cross section as a thin slice of the object. For Example:
Finding Volumes. In General: Vertical Cut:Horizontal Cut:
Section 1.2 – 1.3 Outline Intersection Disjoint Sets (A B= ) AND Union OR Universe The set of items that are possible for membership Venn Diagrams.
Objective: Review. Warm up 1.. Cross-section: It is the intersection of a solid and a plane.
A k = area of k th rectangle, f(c k ) – g(c k ) = height, x k = width. 6.1 Area between two curves.
Y – Intercept of a Line The y – intercept of a line is the point where the line intersects or “cuts through” the y – axis.
You probably know the ellipse by its more well-known name: the oval. An ellipse looks like a circle that has been stretched out.
MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.1 – Area Between Two Curves Copyright © 2006 by Ron Wallace, all.
What you see when you slice.
Vocabulary A polyhedron is a three-dimensional solid with flat surfaces and straight edges. Each polygon is a face of the polyhedron. An edge is a segment.
Volume of Cross-Sectional Solids
Section 6.1 – Volume using Cross-sections – Disks and Washers Copyright 2010 Pearson Education, Inc. Volume of the disk r.
The Miracle of Knot 1.Knot Theory 2.Tricolorability.
Scanning Confocal Basics n What’s so special ? n Confocals give higher resolution than normal microscopes. n Confocals can OPTICALLY SECTION a sample.
Cylinders and Quadric Surfaces
6.2 - Volumes Roshan. What is Volume? What do we mean by the volume of a solid? How do we know that the volume of a sphere of radius r is 4πr 3 /3 ? How.
Created by Mandy Plunkett Modified by Charlotte Stripling
© 2017 SlidePlayer.com Inc. All rights reserved.