Special spines of 2-fold branched coverings of 3-sphere
by
O. M. Davydov
Chelyabinsk State Agricultural University

We suggest a way to construct special spines of 2-fold coverings of S³ branched over links. A similar result was obtained by M. Ferri ([1]) in terms of crystallizations.

Let L be a link. The 2-fold covering of S³ branched over L is S³ iff L is trivial. Given non-trivial L, we construct the following graph. For each double point of a planar projection F of L we draw a circle around it. The intersection points of circles and the link are vertices. Two vertices are symmetric, if they are joined by an overpass of F. If a vertex is not contained in an overpass, we say that it is symmetric with itself. We erase interiors of circles. Now the segments joining vertices are edges (that are parts of circles and F). We join a pair of vertices by an additional edge iff their symmetric ones are joined by an edge obtained from the part of F. We obtain an regular graph G of valency 4.

Denote by \Gamma₁ the graph which constructed from G by removing the edges obtained from parts of F, and by G₂ the graph which constructed from G by removing the additional edges. We assume that G₁ and G₂ are embedded to the plane without self-crossings.

Now we construct the special 2-polyhedron P as follows:

(a) attach a unique 2-cell to each circle;

(b) attach a unique 2-cell to each cycle which consists of edges obtained from parts of F and additional edges;

(c) attach a unique 2-cell for each cycle which consists of edges that are obtained from circles and those obtained from parts of F such that the corresponding cycle in G₂ bounds a plane polygon;

(d) attach a unique 2-cell for each cycle which consists of edges that are obtained from circles and additional edges such that the corresponding cycle in G₁ bounds a plane polygon.

Theorem. P is a special spine of the 2-fold covering of S³ branched over L, and G is the singular graph of P.

Using the method of [2, 3] we enumerate and recognize all 2-fold coverings of S³ branched over simple knots up to 10 crossings. The result for knots up to 8 crossings is presented below. We use Rolfsen notation. Fib₄ means a Fibonacci manifold H³ / < x₁, ... , x₄ | x_i x_i+1 = x_i+2, i \mod 4 > of complexity 11 ([4]).By M₁, M₂, M₃ we denote manifolds of complexity 8 unrecognized by the author. According to the theorem from [2] any such manifold has a Seifert structure. Other manifolds in the table have complexity less than 7.

REFERENCES.

1. Ferri M., Crystallisations of 2-fold branched coverings of S³, Proc. of Amer. Math. Soc., 72 (1979), 271-276.

2. Matveev S. V., Complexity theory of 3-manifolds, Acta Applicandae Mathematicae, 19(1990), 101-130.

3. Matveev S. V., Tables of 3-manifolds up to complexity 6, Bonn, MPIM, Preprint Series, 67, 1998.

4. Mednykh A.D., Vesnin A. Ju., The geometry and topology of the Fibonacci manifolds, Preprint, 1995.

knot	covering	knot	covering	knot	covering
3.1	L3, 1	4.1	L5, 2	5.1	L5, 1
5.2	L7, 2	6.1	L9, 2	6.2	L11, 3
6.3	L13, 5	7.1	L7, 1	7.2	L11, 2
7.3	L13, 3	7.4	L15, 4	7.5	L17, 5
7.6	L19, 7	7.7	L21, 8	8.1	L13, 2
8.2	L17, 3	8.3	L17, 4	8.4	L19, 4
8.5	S3/(P4 ×Z7)	8.6	L23, 7	8.7	L23, 3
8.8	L25, 9	8.9	L25, 7	8.10	S3/P216
8.11	L27, 8	8.12	L29, 12	8.13	L29, 8
8.14	L31, 12	8.15	M1	8.16	M2
8.17	M3	8.18	Fib4	8.19	S3/P24
8.20	S3/P72	8.21	S3/(P24 ×Z5)

Date received: May 25, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadc-02.