Moving Knots and Knotted Surfaces

This short article explains my joint research with J. Scott Carter and Joachim Rieger, \A combinatorial description of knotted surfaces and their isotopies" 3]. For other topics in knotted surface theory that I am interested, see the book with J. Scott Carter \Knotted Surfaces and Their Diagrams" 4]. Please also check publication list in my home page for more details. This article is intended for people with little background in Mathematics or Knot Theory. In the above mentioned paper we provided new methods for the study of knotted surfaces in the 4-dimensional space by visual-izations. The methods we proposed include movie descriptions where we regard the 4-dimensional space as the space-time, and projections of knotted surfaces into the 3-space where they can be actually visualized.