For knots with 10 or fewer crossings, we use the classical names, as tabulated for instance by Rolfsen, eliminating the duplicate 10162 from the count. For 11 crossing knots, we use the Dowker-Thistlethwaite name convention, based on the lexigraphical ordering of the minimal Dowker notation for each knot.
There is a check box to include the Dowker-Thistlethwaite names for all knots, as is used by Knotscape. As an aid to moving between the two naming conventions, the "name rank" is also included as an optional checkbox. Thus, to see the table of knots using the DT conventions, in the order of DT, download the table using DT name and rank, and sort by the DT rank.
There was also a set of classical names given to the 11 crossing knots, with additions by Conway and Perko. These will be given under another check box, currently under development.