Width


The width of a regular projection of a knot is the maximum number of points of intersection of the projection and a horizontal line. The width of a knot is the minimum width among all possible projections.

For a two bridge knot the width is clearly four. For a prime three bridge knot, the width is six. (To see this, note that for any projection, the number of intersection points of generic horizontal cross-sections forms a sequence of even integers changing by 2 at each maximimum or minimum. If the the width of a knot is four, the sequence would be of the form 0, 2, 4, 2, 4, 2, . . . , 4, 2, 0. If any 2 appears between two 4s, then primeness would permit the diagram to be simplified without increasing the width. Thus, the sequence must be of the form 0, 2, 4, 2, 1, and the knot would be two bridge.

Of the 801 prime knots of 11 or fewer crossings, 186 are two-bridge and 600 are three-bridge. The remaining knots are: 11a_43, 11a_44, 11a_47, 11a_57, 11a_231 11a_263, 11n_71, 11n_72, 11n_73, 11n_74, 11n_75, 11n_76, 11n_77, 11n_78, 11n_81. All can be seen to be of width 4 by inspection.


References