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6.1 A comment on Figure 1

It would be worthwhile for the reader to reflect on the relationship between the relations and the syzygies of $ B_4$ and singularities of plane curves. One such codimension one singularity is the triple point \includegraphics[width=0.5in]{figs/TriplePoint.eps}, which corresponds to the last two relations above, which can be viewed as ``the motion of a double point across a line''. One such codimension two singularity is the quadruple point \includegraphics[width=0.7in]{figs/QuadruplePoint.eps}, and it corresponds to the syzygy of Figure 1: There is a circle-worth of generic deformations of the quadruple point, corresponding to ``the cross rotating around the target'': \includegraphics[width=0.7in]{figs/CircleWorth.eps}. The different codimension one singularities along this rotation are exactly the relations in our syzygy.

Dror Bar-Natan 2001-07-23