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There is an alternative version
here . This version concentrates on the Seifert surface which is visible in gray. The red picture (and the cover illustration of this page) is obtained from this braid picture by means of a 3-dimensional closure process. In the blue version, each 2-dimensional slice is a Kamada braid chart; it is the 2-d slice of the Seifert surface. Most consist of four black vertices (drawn in white or grey) which are the cross-sectional planes intersecting the knot. Choose a particular 2-dimensional level, then slice in the 1-dimensional projection; you'll see a (fairly boring) sequence of braid pictures. When there are four black vertices, the closure of the 2-d braid chart yields a torus embedded in 4-space as a 2-fold branch cover of the 2-sphere. The knotted portion of the 3-d projection describes a sequence of Dehn twists on this torus.
This work sheet illustrates an explicit embedding of the 3-fold branched cover of the 3-ball with the branched set being the trefoil knot. In the projection, you can see the outline of the trefoil. After viewing this look at the 2-dimensional projections/cross sections. Each yields a 2-chart in the sense of Kamada. Each such chart describes a sequence of braid words, and the sequences of charts are connected by either chart moves, or by handle attachments.
In my experience, many viewers have a little trouble with the closures of such figures. This work sheet illustrates the braid closure of the previous example. It is an embedding of the 3-fold simple branch cover of the 3-sphere where the branch set is, again, the trefoil. The coving space is a 3-sphere since the trefoil is a 2-bridge knot, and it is embedded in the 5-ball in such a way that the projection onto the 3-sphere is the branched covering space.
This work sheet is perhaps the most complicated. It illustrates the 2-fold branch cover of a 4-ball branched along a spun trefoil. It is constructed by finding Seifert surfaces for each cross-section in a movie of the knotted surface. Each Seifert surface is a sequence of braid charts for the corresponding cross section, and these are interconnected by isotopy moves or handle attachments. The 2-dimensional cross-sections thereof are Kamada charts. A Kamada chart describes a braid movie and consequently a branched cover of the 2-sphere. These can be closed following the examples above, to give an embedding of the 2-fold branch cover of the 4-sphere branched over the spun trefoil that is embedded in 6-space in such a way that the projection onto the 4-sphere is the covering map. Later, I'll demonstrate an immersion of 3-fold branched cover of the 4-sphere whose branched set is the 2-twist spun trefoil.
This page is an outline of how to work with a classical knot template written in globular. My hope is that it will appear on a bigger blog in the near future.
This page is an outline of how to work with a movie move written in globular. This too should be a one of the bigger blogs soon.