Apparently, every 2-bridge knot can be drawn such that of the four strands in the braid word, one strand remains straightened and is not crossing any of the other strands. Is there a general algorithm on how to proceed? (say you start with any diagram of any 2-bridge knot, and want to achieve such a "standard form", illustrated e.g. in the following picture)
Context: Horst Schubert, classification of rational tangles (1956), e.g. here
Context: Charles Livingston. Knot Theory.
A quick way to prove this is to observe that if you have the LHS in the following picture, you can get to the RHS by dragging the rightmost part of the strand down and around the rest of the diagram.