Back to Scott Carter's home page

Here is a compendium of worksheets in globular that I have been working upon. These are sufficiently complete to be available publicly.

The views and opinions expressed in these web page(s) are strictly those of the author. The contents of these page(s) have not been reviewed or approved by the University of South Alabama. |

This page is best viewed in google chrome. Globular is written with the chrome browser (not to be confused with Chrome fenders ) in mind.

In the process, I noticed that after introducing the two swallow-tails and rearranging the double points that lie along the inner-fold, that there was an unknotted middle cross-section. I introduced an unknotting sequence of Reidemeister moves (and their inverses) to find two unknotted spheres. That these are unknotted follows from an old Theorem of Scharlemann that a knotted sphere in 4-space that has four critical points is knotted. The philosophy of unknotting them is to cancel a saddle/crotch against a death/birth and gradually removing the double curves.

The important parts of the computation involve a number of steps in which Maximal/minimal/Maximal points are brought together to form a valley. Then a beak-to-beak move provides a new connection in the fold set. Often swallow-tails are used.

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.

Next up:

Back to Scott Carter's home page