Do Reidemeister moves allow you to go from a prime knot of n crossings to another prime knot of n crossings? I am trying to go through a set of Reidemeister moves that will go from knot $7(2)$ to knot $7(1)$. I am having trouble getting from one to another and was wondering if it is my lack of creativity or it is just not possible. 
*Using Alexander Notation
 A: Reidemeister proved that two knots are equivalent if and only if they can be connected through a sequence of his three eponymous moves.
One hint there is no sequence of moves connecting $7_1$ to $7_2$ is that Alexander-Briggs notation has it where two knots are given the same name if and only if they are equivalent.  (The first number is the minimal number of crossings in any diagram of that knot, and the second number just indexes where it appears in that part of the knot table.)
A way to see the two knots are not equivalent is to calculate some sort of invariant.  The knot determinant is not too difficult to calculate but takes some theory to understand (they have determinants of $7$ and $11$, respectively).  The have different Jones polynomials, which are easier to learn how to compute.
It's easy to see the unknotting number of $7_2$ is $1$, and, while this doesn't prove anything, it's hard to imagine the unknotting number of $7_1$ is any less than $2$ (and its unknotting number is in fact $3$).
http://katlas.org/wiki/7_1
http://katlas.org/wiki/7_2
