Surgery presentation of a manifold

Assume that you have a connected planar diagramm of a framed link L with n crossings. In order to write down the presentation accepted by the program, do the following (steps 3 and 4 are optional):

  1. Number the crossings of L by successive natural numbers 1,2,3,...,n in an arbitrary manner.
  2. Choose an orientation and an initial point for each component of L.
  3. Assign orientation signs +1 or -1 to each crossing using the convention shown in the figure below: we take +1 if the overcrossing strand passes the undercrossing one from the left to the right, and -1 otherwise.
              ^                            ^
              |                            |  
              |                            |
         ---------->   +1             <----------   -1  
              |                            |
              |                            |
                               
    
  4. Write down a string of n integers +1 or -1: i-th number of the string is the orientation sign of i-th crossing.
  5. Order the components of L in some arbitrary manner.
  6. Start “walking” along the first component of L from the initial point, taking note of the numbers of the crossings you’ve gone through. If in a given crossing labeled with a number k you are on the “over” strand, write down k or +k. If you are on the “under” strand, write down -k.
  7. Do the same for all other components of L (if any). The resulting list of signed integers (in the case of a knot) or list of lists of signed integers (in the case of a link) is called the Gauss Code of L.

Examples.

1. Figure-eight knot without framing

2. Trefoil knot with framing -3/2

3. Hopf link with framings -2 and 5/7

4. Whitehad link with the secondd component framed

Remarks.