A collection of globular worksheets relating to knotted surfaces and braided manifolds

December 15, 2016 A lens space

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Here is a compendium of worksheets in globular that I have been working upon. These are sufficiently complete to be available publicly.
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  • This work sheet first appeared at the n-lab in the summer of 2016 after I began several conversations with Jamie Vicary about how to manipulate knotted surfaces in 4-dimensional space. It is an example of the 1-twist-spun trefoil. The sphere is known to be knotted via Zeeman's theorem. On the other hand, geometric proofs are desirable. In our book Knotted surfaces and their diagrams Masahico Saito and I gave a diagrammatic proof in the case of 1-twist spinning the trefoil. This work sheet is based upon that calculation.

    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.

  • This work sheet illustrates (in far too many steps) how to go from Kamada's braid form of the spun-trefoil to a standard diagram thereof. Here Kamada's black vertices are illustrated in white (my bad), and they are replaced by their generic singularities. I call the generic pictures chevrons since that is their shape in the more realistic depiction.

    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 2-fold branched covering of the 3-sphere branched over the trefoil.
  • 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 the chart moves in the sense of Kamada. The CIII-moves that are described as "pushing a black vertex through a crossing" follow from naturality of crossings (distant crossings commute), and so are not listed. Meanwhile, there are 48 variations of pushing a black vertex through a white vertex that depend upon the signs of the incoming strings at the type III move, top/middle or middle/bottom, incoming/outgoing, and direction 121=>212 or 212=>121 of the type III move. This worksheet is a utility work sheet. In the next few, I'll used these cells to create embeddings of branched covers of the 3-sphere (or 4-sphere) branched over a given knot (or knotted surface). There is a bullet item below for sample proofs. That will be filled in after more examples are created.
  • 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:

  • Sample proofs of the CIII moves

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