the knot surgery - from a $6^3_2$ knot to a $3_1$ trefoil knot It is intuitive that one can simply doing a cut-gluing surgery to make a $6^3_2$ to a $3_1$ trefoil knot:
e.g. from

to

All one needs to do it to cut the three intersections at the angle of $\pi/6$, $\pi/6+2\pi/3$, $\pi/6+4\pi/3$ and then gluing three intersections.

question: So what is the precise mathematical formulation of this procedure? And how to write a math formula to implement this procedure?

i.e. What I was hoping to know is something like
$$
3_1= \text{function[cut and glue]}\circ [6^3_2]
$$
 A: You are probably looking for skein relation that features prominently in the definitions of Conway, Jones, and HOMFLY polynomials. Notice that skein relation includes not 2 but 3 links: knot invariants are usually defined as relations between all 3 knots/links, not just 2 of them.
A: There is a type of "move" called an $n$-move, which takes two strands which run parallel and places $n$ positive half twists into them.  These kinds of moves, along with the normal Reidemeister moves, are usually studying in if they can unknot a knot. For example, it is not hard to realize that the $1$-move is an unknotting operation, since we can always just take out any crossing with it.
The only reference I could think of quickly was Przytycki's paper which defines them (definition 1.8) and how they relate to coloring.  I am sure there are a lot of papers that deal with them in general, that you can find, now that you know the name of such a move.
I am not sure how often people try to write a formula or equation for such things though.
