MorseLink Presentations: Difference between revisions
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*'''Cap[m,n]''' is an annihilation of the strands m and n, again with m and n differing by 1, and is directed from m to n. |
*'''Cap[m,n]''' is an annihilation of the strands m and n, again with m and n differing by 1, and is directed from m to n. |
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*'''X[n, d, a, b]''' is a crossing of the n-th and the (n+1)-th strands. 'd' signifies whether the n-th strand goes 'Over' or 'Under' the (n+1)-th strand. 'a' indicates whether the n-th strand is moving 'Up' or 'Down' through the crossing, relative to the chosen axis; 'b' does the same for the (n+1)-th strand. |
*'''X[n, d, a, b]''' is a crossing of the n-th and the (n+1)-th strands. 'd' signifies whether the n-th strand goes 'Over' or 'Under' the (n+1)-th strand. 'a' indicates whether the n-th strand is moving 'Up' or 'Down' through the crossing, relative to the chosen axis; 'b' does the same for the (n+1)-th strand. |
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For concreteness, let us find the MorseLink presentation of the following picture of the knot [[4_1]]: |
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<!--$$<<KnotTheory`$$--><!--$$Show[DrawMorseLink[Knot[4,1]]]$$--><!--END--> |
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Revision as of 19:35, 1 September 2005
The MorseLink presentation describes an oriented knot or a 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 (resp.) relative to the chosen axis. The conventions used by MorseLink are as follows:
- Cup[m,n] is 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] is 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 n-th and the (n+1)-th strands. 'd' signifies whether the n-th strand goes 'Over' or 'Under' the (n+1)-th strand. 'a' indicates whether the n-th strand is moving 'Up' or 'Down' through the crossing, relative to the chosen axis; 'b' does the same for the (n+1)-th strand.
