MorseLink Presentations
The MorseLink presentation describes an oriented knot or link diagram as a sequence of 'events'. To begin, we fix an axis in the plane, so that each event in the MorseLink description occurs in successive intervals along this axis. The 'events' that comprise a knot or link are 'cups' (creations), 'caps' (annihilations), and crossings, where cups and caps are the minima and maxima (respectively) relative to the chosen axis. The conventions used by MorseLink are as follows:
 Cup[m,n] represents a creation, where the new strands are in position m and n, with m and n differing by 1, and is directed from m to n.
 Cap[m,n] represents an annihilation of the strands m and n, again with m and n differing by 1, and is directed from m to n.
 X[n, d, a, b] is a crossing of the nth and the (n+1)th strands. 'd' signifies whether the nth strand goes 'Over' or 'Under' the (n+1)th strand. 'a' indicates whether the nth strand is moving 'Up' or 'Down' through the crossing, relative to the chosen axis; 'b' does the same for the (n+1)th strand.
For concreteness, let us find a MorseLink presentation of the knot 4_1, based on the following diagram:
We fix the chosen axis to be the positive axis, and we number the strands at each interval from bottom to top. Clearly the first event to take place is a cup, creating strands 1 and 2, and directed from 1 to 2. The next event is also a cup; this time, the strands 3 and 4 are created, from 3 to 4. So the first two elements of our MorseLink presentation are {Cup[1,2], Cup[3,4], ...}
.
The next event to take place is a crossing. This crossing has strand 2 going under strand 3; following the orientations, strand 2 moves to the right through the crossing, while strand 3 comes from the left. Since the chosen axis is pointing to the right, this event is described as X[2, Under, Up, Down]
. We may proceed in a similar way for the rest of the crossings.
After three more crossings, we encounter two cap events. The first annihilates strands 2 and 3, and is directed from 2 to 3. The second annihilates strands 1 and 2, but is directed from 2 to 1. So the MorseLink presentation for the knot 4_1 is given by: MorseLink[Cup[1,2], Cup[3,4], X[2 , Under, Up, Down], X[2, Under, Down, Up], X[1, Over, Down, Up], X[1, Over, Up, Down], Cap[2,3], Cap[2,1]]
.
Further notes
KnotTheory` knows about Morse link presentations: (For In[1] see Setup)


 Obviously, there is no unique Morse link presentation for a knot. One presentation may be considered 'better' than another if it generally has fewer strands. Unfortunately, this program makes only a halfhearted attempt in coming up with such presentations; it does a creation only when it can no longer do any caps or crossings, but exhibits a lack of foresight into which edges to create.