The site KnotInfo can classify many small knots up to mirroring given a Dowker–Thistlethwaite Code of the knot, and can even identify whether a knot is a connected sum of other small knots.
Being very inexperienced in knot theory, I don't understand how exactly it does this. I had thought there might be a way to manipulate the DT code in a way reminiscent of Reidemeister moves in order to simplify the diagram, but I haven't been able to find any online resources that suggest how this might be done.
In particular I'd appreciate an explanation of the algorithm, and/or a link to a resource describing it.
Thanks