The foundations of mathematical knot theory, in its classical form, lie in the late 19th century. Classic mathematical knot theory is concerned primarily with the study of the topological properties of idealised knots (a mathematical knot is "a homeomorphism that maps a circle into three dimensional space and cannot be reduced to the unknot by an ambient isotopy"). Such a theory is relatively remote from the everyday activities of people who use knots professionaly or in their leisure activities.
The more recent branch of knot theory, ideal knots, is concerned with the study of "minimal length trajectories of uniform diameter tubes forming a given type of knot". It has applications to problems in biology and chemistry, specifically to understanding the structure of DNA and other macromolecules such proteins and polymers, and to fundamental issues in theoretical physics. As such it is arguably less divorced from reality but still relatively remote from the everyday.
In contrast the study of "real knots" is concerned with knots which can be tied in real physical lengths of rope. Topics of interest include the classification of knots, assessment of the strength and security of knots, mechanisms for the failure of knots, the symmetries of knots and their effects on knot properties, and techniques for tying knots. One particular goal, so far unresolved, is the devising of a sound mathematical notation for representing methods of tying knots.
I have been involved in teaching knots and ropework to novice seamen for some 20 years and, as an academic engineer and mathematician, had always been concerned (as an academic inevitably would be!) that there was no sound fundamental theory of such knots, only a widely accepted corpus of empirical knowledge and professional practice. That the practices were so widespread and generally agreed suggests that there must be a sound basis, but the theory of that basis is notably absent. Around 2000 I came across Roger Miles' book "Symmetric Bends" (Symmetric Bends: How to Join Two Lengths of Cord, World Scientific, 1995, 9810221940) and was inspired to try to develop his work and make some contributions to a putative theory of real knots. And that's how it all started ...
Here are some short articles (linked to full text pdf files) which have been published in Knotting Matters, the journal of the International Guild of Knot Tyers.