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# 数学代写|低维拓扑代写Low Dimensional Topology代考|MATH145B Branched Along a Knotted or Linked Surface

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## 数学代写|低维拓扑代写Low Dimensional Topology代考|The 2-fold Branched Cover of S4 Branched Along a Knotted or Linked Surface

In [2] a Seifert algorithm for knotted surfaces that project without branch points is presented. This algorithm was adjusted by the second author in [9] to be applied to the surface braid case. In our respective books, [3] or $[10]$ these algorithms are described in detail. The 2-fold branched cover of $S^4$ branched along an orientable knotted or linked surface can be constructed from two copies of $S^4$ both cut along the Seifert solid, and gluing the positive side of one copy to the negative side of the other.

The braid chart can be used to give a movie description of the branch locus. In such a movie, or in a standard movie description of the knotted surface, the Seifert solid is cut into 2-dimensional slices that are Seifert surfaces for the classical cross-sectional links. Thus for any such cross-section, the previous section provided an embedding of the 2-fold branched cover of that cross-section. These are, then, connected by either handle attachments or isotopies that are induced by the Reidemeister moves.

Specifically, the birth of a simple closed curve corresponds to a 1-handle attached between the successive 3-dimensional 2-fold branched covers. A 1-handle attached between successive movie stills corresponds to a 2-handle attached between successive 3-dimensional 2-fold branched covers.

Following the discussion of the 3 -fold branched covers of classical knots, an example of 3 -fold branched cover of the 4-sphere branched along the spun trefoil will be given. These examples are more complicated than the 2-fold branched coverings.

In Fig. 25 we indicate that the 2-fold branched cover of $S^4$ branched along an unknotted sphere is $S^4$, and we indicate an embedded folding. Meanwhile, the 2-fold branched cover of $S^4$ branched along an unknotted torus is $S^2 \times S^2$ and also has an embedded folding.

## 数学代写|低维拓扑代写Low Dimensional Topology代考|Three-fold Simple Branched Covers

In this section, we construct embeddings and immersions of 3 -fold branch covers of the 3- and 4-sphere branched over the trefoil, the knot $7_4$, and the knotted sphere that is the spin of the trefoil. We begin with the trefoil as indicated Fig. 26. The trefoil is 3-colorable, and we can lift this coloring to a representation to the 3-string braid group (also the fundamental group of the knot itself) by assigning the braid generator $\sigma_1$ to the (blue) label 1, assigning the braid generator $\sigma_2$ to the (red) label 2, and assigning $\sigma_2^{-1} \sigma_1 \sigma_2$ to the (green) label 3. From this coloring, a sequence of braid charts is constructed. The endpoints of the arcs in the charts correspond to black vertices, and as the charts are stacked, these end points trace the outline of the knot. Successive braid charts differ by a planar isotopy, by one of the chart moves, or by the addition or subtraction of a simple arc. As before, the addition or subtraction of an arc corresponds to attaching a handle between successive sheets in the covering. The sequence of charts can be interpolated in 3-space to create a (non-generically) immersed surface in 3-space that we call a curtain. To construct the 3 -fold branch cover, we take three copies of $S^3$, cut each along the appropriately labeled sheets of the curtains, and re-glue. The embedding in this case is achieved by interpreting each chart as a braided surface and interpolating between successive surfaces.

## MATLAB代写

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