Is it known which quandles arise as the knot quandle of some knot? I seem - vaguely - to remember hearing that all (finite?) quandles arise this way but I've failed to find a reference to this fact.

Personal anecdote, John Stallings showed me the initial knot quandle paper, showing that the quandle did a wonderful job of classifying knots. His comment was that there were plentiful imponderables, such as deciding whether a prescribed quandle was trivial.
– Will JagyNov 3 '10 at 17:25

what is a quandle?
– Paolo AcetoNov 3 '10 at 18:14

@Will-Jagy, so a quandle has to have both the right-distributive and the left-distributive properties of a rack, along with having $a\triangleright a=a$ and $a \triangleleft a = a$? Is the matrix-form the best way to represent it? (sorry to be asking another question as a comment...)
– sleepless in beantownNov 7 '10 at 3:48

1

Each knot quandle is connected.
– Zhiyun ChengApr 23 '14 at 9:33