Picturing twisting of strands explicitly after blow downs In order to simplify Kirby calculus proofs, one can use a box notation which indicates a number of full twists up to sign. In the scenario of two strands with twist boxes, it is straightforward to draw the corresponding picture.
Let's assume that we have three strands with -3 and 4 full twists. I am very confused when I have tried to draw these diagrams.
How can we draw them explicitly? What is the algorithm? Any reference for explicit drawings would be useful.
 A: Okay, here are some pictures.  You should imagine these as strands on a tube.  A full twist takes the one end and rotates it $2\pi$ leaving the other fixed and everything smoothly twists along the tube.  Here it is done with one strand and then 3 strands.  Then I make the 3 strands a little cleaner and last I turn it into a diagrammatic picture.  Which you can think of as a braid with braid word $\sigma_1\sigma_2\sigma_1\sigma_2\sigma_1\sigma_2$, reading from the top down. Or the same word with inverses on everything, depending on convention or direction of your rotation around the tube.  For more strands, it is just the braid word for the torus knot with that many strands.
This will be one full twist.  For 4 twists, you just stack 4 of these on top of each other.  And for -3 twists, it twists in the opposite direction and then stack 3 of them on top of each other.
Disclaimer!  I did not look up what is standard for Kirby diagrams, so I might have just drawn a $-1$ twist instead of a $1$ twist.  Please double check what you should be using or you might end up with a sign problem in whatever you are computing.


 
 
