Knot Theory: Smooth vs Polygonal So I will be giving a talk about knot theory and was wondering why would one study knots from a graph theoretical perspective, i.e a collection of edges and vertices? 
Is this just a preference? Is there a limit to the amount of information one can get from a collection of polygonal lines versus a smooth curve?
 A: In knot theory one studies "isotopy classes" of knots, which, intuitively, means that knots are regarded as the same if one can be deformed to the other without introducing self-crossings (the actual definition is more complex). Polygonal knots are easy to grasp and they are computer-friendly, since describing such a knot amounts to prescribing finitely many sets of coordinates. Take a look 
here and here for real implementations. 
On the other hand, using smooth knots more intuitive (your shoe strings or knots used by sailors and others more resemble smooth knots than polygonal ones!). However, one can always replace a smooth knot with polygonal one and vice versa without changing its isotopy class. I think, if you pick any book about knots, it will explain why this is the case. (Try "The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots" by Colin Adams; see also here for further suggestions.) Intuitively, you approximate a smooth function by piecewise-linear one, conversely, given a polygonal knot you "smooth the corners" to make a smooth knot.  
See also this and this MSE posts on this two-way relation. 
In either setting, to recover complete info about the isotopy class of a knot, it suffices to draw its diagram on the plane (smooth or polygonal, does not matter) and prescribe which intersections are overcrossings and which are undercrossings.  
