In a math textbook and this article in NRICH, some problems deal with a special kind of knots: those which are formed from several strands:
The problems ask if a given knot can be formed from just one strand, and if not, how many strands are needed.
What is the mathematics behind these problems?
As the article you link mentions, these problems pertain to the area of mathematical study known as Knot Theory. If you are really interested in the mathematics behind the study of knots (since the article specifically mentions these problems are selected so as not to require any specific mathematics), I would recommend picking up The Knot Book, by Colin Adams. It's a highly readable introduction to knot theory that does not require any advanced mathematical background. If you are interested in some more advanced mathematics, I will simply mention that knot theory is usually set in the context of Topology. Combinatorics plays a heavy role, too. You will definitely see quite a bit of that in the Wikipedia article.
Additionally, as is mentioned in the comments, a "knot" which is composed of multiple "strands" is known as a link. Asking whether a link as shown in the diagram "can be formed from just one strand" seems a little odd, as you would just follow each strand around until it loops back and count the number of strands. The number of strands you count is the number of strands it can be made from, that's it. At any rate, the background information you're looking for will be found in the references mentioned above. I hope that satisfies your curiosity regarding knots.