This is a fascinating area of research as it initially seems so simple and then unravels in layers of complexity which were totally unexpected. For instance, I've developed an algorithm to reduce a braid to its minimum length in terms of its crossings. This paper is currently being considered for publication in J. Phys. The same aim was the subject of an elastic string simulation which I carried out and it was discovered that elastic strings do not reduce the crossing number to anywhere near its minimum through reducing the energy. Download this paper here .
Knots are difficult to work with because they are topological objects.
To reduce this complication, I've developed a comprehensive notation for knots which allows a much neater representation and simplifies certain calculations. To this end I have developed an algorithm which turns a knot into a closed braid. This algorithm provides the needed step between knot and braid theory. Essentially all of knot theory can be done via braids because all knots can be represented via a closed braid. There have been other such methods, but this is the most efficient one in that the final braid is the smallest in the number of crossings and strings used.