I'm reading Kauffman's 1990 paper "An Invariant of Regular Isotopy" about knots that are isotopic through only Reidemeister Type II and III moves, which is known as a regular isotopy. His paper claims there is a relationship between regular isotopy and embedded bands ($S^1 \times [0,1]$) in $S^3$. He refers to Burde's Knots textbook, but I can't find any mention of regular isotopy, because it seems Kauffman coined that phrase in the paper, after Burde's text was written.


I think a regular isotopy of knots corresponds to an embedded band. However, I'm worried there might be a pathology I'm overlooking. Does anyone have a precise statement of the relationship?



Embedded ribbons and regular isotopy
From any knot diagram, one can obtain a framed knot by taking the "blackboard framing." The point of regular isotopy of knot diagrams is that it preserves this blackboard framing. Since framed knots and embedded bands are the same thing, regular isotopy will also preserve the embedded band corresponding to the blackboard framing of the knot diagram.


I assume this is discussed in more detail in Burde, maybe in terms of framed knots. It's also possible Burde doesn't discuss framed knots at all, since I think people became much more interested them after the discovery of the Jones polynomial/the Chern-Simons TQFT. And I agree: I think Kauffman coined the term "regular isotopy," so it is probably not used in Burde.


