Does any proof of the knottedness of the trefoil capture our intuition? The fact that trefoil knot is knotted, i.e. not equivalent to the trivial knot, was first proved in the early 20th century by Wirtinger and Tietze.  They used knot groups, though there are now 
countless other methods to do it, like Fox 3-colorings and the Jones polynomial. 
But I think it's fair to say that humans have intuitively believed that the trefoil is knotted approximately forever.  I also think it's fair to say that the intuition behind our belief is neither based on knot groups, nor Fox 3-colorings, or even the Jones polynomial.  It's based on the notion that however you move a trefoil around, it's never going to become a trivial knot, a notion that is clear to anyone who plays around with a trefoil.
So my question is, is there any proof of the fact that the trefoil is knotted that captures this intuition?  Perhaps a direct proof that no ambient isotopy changes the trefoil into a trivial knot, rather than a proof that resorts to knot invariants?  Our does our intuition bear absolutely no relation to the actual reason why the trefoil is knotted?
 A: I first want to express my hesitation at answering the question the way you pose it here.  Intuition is extremely valuable, but it can lead you astray if you are not careful.  Nonetheless, I understand that you want an intuitive proof that shows the trefoil is not the unknot.  
The problem comes from the fact that there is no way, in general, to be sure we have "simplified a knot all the way." But the trefoil is an alternating knot, which is a special kind of knot.  Tait conjectured:

Reduced alternating diagrams have minimal link crossing number.

And this was proven later by Kauffman, Murasugi, and Thisthlethwaite using the Jones polynomial.  This is a highly non-trivial proof, but the thing you want is in some way contained in that conjecture.  So find a three crossing diagram of the trefoil and notice it alternates and you are done!
Now, I want to point out that this is not proven for non-alternating knots, because it is not true for non-alternating knots.  You can almost surely prove any individual knot you come across is non-trivial with some work, but knots can be trivial and not look that way at all.  I will just reference this post on MathOverflow for some very unfriendly unknots that they have pictured there, so I will not repost them here.
A: I think the intuitive proof runs into trouble because we want to show that no ambient isotopy changing the trefoil to the unknot. You can prove that two knots are equivalent by exhibiting a specific isotopy and this is very intuitive. But if you want a rigorous proof that no isotopy exists you usually need some extra property to set up a proof by contradiction. So any rigorous proof essentially says, if there where an isotopy between the trefoil and the unknot that this isotopy would preserve property X and property X is different for the trefoil and the unknot, therefore no isotopy exists. At best you could find a property X that is easy to compute and and where it is easy to see that it is preserved by isotopies.
A: I like my own "true at a glance" picture-proof that the linking number between branch curves of a trefoil's 3-fold non-cyclic covering space is non-zero. See Figure 5 of my paper "On Covering Spaces of Knots" [Glasnik Mat. 29 (1974), 141-145].
It's 100% intuition, dating back to the illustration in Heegaard's 1989 thesis for singularities of the complex function x^2-y^3, and about as simple as such matters get.
For hundreds of more complicated examples see the project "On 3-colored knots" on my Researchgate site. --Ken Perko, lbrtpl@gmail.com.
