Devise a method for constructing a table of knots, and use it to find $10$ knots of not more than $6$ crossings (do not consider the question of whether these are really distinct types.)
Could anyone give me a hint for answering this question, please?
On
The unknot gives you one knot to start with. Now use the process shown below to construct knots with $\geq 3$ crossings.
Up to six crossings, this gives four more knots (because the first valid knot using the method is the trefoil. Now consider the mirror images of the knots constructed above; this gives another four. We now have nine knots in our table.
Since we don't care about equivalence, one could perhaps change the diagram a little so there are two crossings at the top, as opposed to one, and add crossings to the bottom as before. Since you only need ten knots, we'd only have to do this once.
Since you do not have to pay attention to potentially constructing equivalent knots, you can just do that by the number of total crossings. Start with $0$ crossings, i.e the unknot. Then you can add one crossing etc. Once you have some knots you can also use mirror images to get more. That actually also leads to non-equivalent knots as for the tre-foil knot.