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):
- Number the crossings of L by successive natural numbers
1,2,3,...,n in an arbitrary manner.
- Choose an orientation and an initial point for each component of L.
- 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
| |
| |
- Write down a string of n integers +1 or -1: i-th number of
the string is the orientation sign of i-th crossing.
- Order the components of L in some arbitrary manner.
- 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.
- 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.
- The description of a presentation by a surgery starts with the command link
and ends with the command end.
- After the command link, you may put optional command
signs followed by the string of integers produced at Step 4 above.
- Then for each component you put one or two commands: code followed by the code of that component
that was obtained at Steps 6-7, and optional framing followed by the numerator and
the denominator (separated by the blank space only, or by the sign "/") of the corresponding component.
If the command framing is omitted corresponding component is not framed.
See the examples below.
Examples.
1. Figure-eight knot without framing
link
code 1 -2 3 -4 2 -1 4 -3
end
2. Trefoil knot with framing -3/2
link
signs 1 1 1
code 1 -2 3 -1 2 -3
framing -3 2
end
3. Hopf link with framings -2 and 5/7
link
signs -1 -1
code -1 2
framing -2
code -2 1
framing 5 7
end
4. Whitehad link with the secondd component framed
link
code -1 2 -3 4
code 1 -5 3 -4 5 -2
framing -3/2
end
Remarks.
- Framings n and n 1 are equivalent.
- If steps 3 and 4 above are omitted but the given code of a knot or link is valid, then
the program calculates some set of signs automaticaly. But this set is only one of many possible ones.
Hence the link that the Recognizer deals with may be different from the one you mean. So it is
recommended to give signs of crossings explicitly and use the command signs. On the other hand,
if you did not do so, you cal always learn which signs were used by the program by going to the menu
"Information|Short".