In a manner similar to the case of knots, a link notation needs to face the same four challenges. As quoted in the discussion of knots, these challenges may be stated as follows.
The answer to this challenge is affirmative. The proof may be summarized as follows. Two link components, say (i) and (j) may indeed be topologically "unable" to "touch" each other and create the "artificial" crossing. But if this is the case, then there do exist other components standing "between" them as a "barrier". Then, one may use a different pairing of link components to create the artificial crossing, and by doing so, the number of remaining components is reduced by one. By continuing this way, one gradually creates all the necessary artificial crossings, until the presentation consists of just one component.
One may argue at this point that while there are always components that may be able to "touch" each other, we do also require that they touch through segments with "parallel" orientations. One may answer this argument by noticing that at the figure below:
A | | C----->------O----->-------D | | B
Let C-O-D and A-O-B be the segments of two distinct link components. If AO and OD have opposite orientations, (i.e. A-O-B points from A to B), then AO is "parallel" to CO and OB is "parallel" to OD. Therefore, in any given link presentation, one may always use the procedure we have presented, in order to reach some notation.
To continue with the discussion of the other four challenges, please click here.
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