pretzel
Welcome to the pretzel page ! Pretzel is a six-minute computer animation created to illustrate the proof of a thoerem about Heegaard splittings of three-manifolds ([Se]).This page contains clips from the video along with a sketch of the proof of the theorem. If you are not interested in the mathematics, feel free to take a look at the pictures and movie clips anyway. For viewing I suggest 'make the eyehook' and 'slide the eyehook'. Let me know what you think !                           
                                                                                                - Eric Sedgwick -

the knot (mpeg 330K)

Here are three different projections of the same pretzel knot, K. Twisting the central region of K will produce projections of K with more crossings. This may be continued to obtain a projection with as many crossing as desired. .

the spines: S1,S2,S3 ... ,(mpegs 660k each)

S1 Associated with each projection is a graph called a spine. Note that:
  • There are an infinite number of spines S1,S2,S3, ...
  • genus = g(Si) = 2i+2.
  • K can be isotoped to lie on each spine Si.
  • Each Si is unknotted. (After performing the isotopy, the complement of the spine union the knot is a handlebody).
  • Each Si is the spine of an incompressible surface (See Parris, [Pa]).

Let M=K(1/n) be a manifold obtained by 1/n Dehn surgery on K. Casson and Gordon noted that each Si will be the spine of a Heegaard splitting of M (The boundary of a regular neighborhood of Siis a surface which decomposes M into two handlebodies) . By applying Casson's rectangle condition they proved:

Theorem (Casson and Gordon, [CG]) Let M = K(1/n) the manifold obtained from 1/n Dehn surgery on K.. If n > 5 then for each i the spine Si is the spine of an irreducible Heegaard splitting of M. In particular, the manifold M has an inifinite number of irreducible Heegaard splittings

(See [Ko] for a related construction). Here we demonstrate a special case of:

Theorem (Sedgwick, [Se]) Let Si and Sj be any pair of the above spines. Then the Heegaard splittings given by Si and Sj are equivalent after one stablization.

We illustrate the proof of this second theorem for the given knot K, the spines S2 and S3 and M = K(1/1). To produce a video of reasonable length the proof is given only for n=1. Repeated application of the same manipulations give the argument for general n.

stabilize (mpeg 2.1M)

Add a trivial handle to the spine S3. Recall that we have performed a 1/n Dehn surgery on the knot K. Therefore a 1/n curve (coordinates in S3) around K bounds a disk in the attached solid torus and represents a trivial handle. (1/n=1/1).

make the eyehook (mpeg 2.1M)

Manipulating the spine by edge slides does not change the Heegaard splitting it represents (it is just an isotopy of the Heegaard surface). Slide one foot of the attached handle along the spine. This converts the trivial handle, a 1/1 curve, into a 1/0 curve (in S3 coordinates). For general n repeat this slide n times..

slide the eyehook (mpeg 629K)

Slide the eyehook into position.

pull a bar out (mpeg 540K)

The eyehook makes it possible to change a crossing between the knot and the spine. Remember, edge slides do not change the spine.

slide the eyehook (mpeg 1.0M)


Move the eyehook to the other side of the knot.

pull another bar out (mpeg 540K)

move the eyehook (mpeg 240K)


Move the eyehook out of the way.

the grand finale - (mpeg 1.8M)

Center the pulled bars, bend the middle bars in, and untwist ......

Note that the final graph contains S2 as a subgraph, and is therefore a stabilization of S2. (For a proof see [Se]). Also, it was constructed by stabilizing S3 once. Thus, the spines S2 and S3 are equivalent after one stabilization. We can also use the same sequence of moves to obtain S1 !

For those of you who would like to have the whole thing: pretzel. (mpeg 16.9M) Watch out, its big !

Credits

pretzel was created and animated by Eric Sedgwick at the Visualization Lab of the Texas Institue for Computational and Applied Mathematics (TICAM). It was created using Maple V.3 and KnotPlot. . Special thanks is due Rob Scharein, the creator of KnotPlot, without whom this would not have been possible.

Please direct any comments to Eric Sedgwick. Click here to email Eric Sedgwick.

References

[CG] Casson, A., Gordon, C., Unpublished.

[Ko] Kobayashi, T., A Construction of 3-Manifolds Whose Homeomorphism Classes of Heegaard Splittings Have Polynomial Growth, Osaka Journal of Mathematics 29 (1992), 653-674.

[Pa] Parris: Pretzel Knot - Ph.D. Thesis, Princetion University (1978).

[Se] Sedgwick, E., An infinite collection of Heegaard splittings that are equivalent after one stabilization. Math. Ann. 308 (1997), no. 1, 65--72.