On normal surfaces in 3-manifolds
by
Evgenii Fominykh
Chelyabinsk State University

Let M be a closed 3-manifold with a fixed singular triangulation T in which there are one vertex and n tetrahedra. An elementary disk in a tetrahedron \sigma is a disk that is properly embedded in \sigma and intersects 2-faces of \sigma in arcs spanning distinct edges of the 2-faces. Two elementary disks belong to the same disk type if there exists an isotopy of M moving one elemementary disk to the other which is invariant on each simplex of T. In each tetrahedron there are three quadrilateral and four triangle disk types.

A normal surface F subset M intersects tetrahedra in elementary disks such that no more than one quadrilateral disk type is represented in each tetrahedron of T. Let \sigma₁, ... , \sigma_n be the tetrahedra of T. For each i, 1 <= i <= n, choose quadrilateral disk type d_i in \sigma_i. Let d=(d₁, ... , d_n). Define a family Q_d of normal surfaces F in M by requiring that any quadrilateral disk in F \cap \sigma_i has the type d_i. Then, to any normal surface F subset Q_d we can assign a vector [x\vec]_F=(x₁, ... , x_n) of nonnegative integers by letting x_i denote the number of elementary disks in F of type d_i. Any such vector [x\vec]_F is a solution to some system S_d of n+1 linear homogeneous equations. Note that a connected normal surface containing only triangle elementary disks is a 2-sphere surrounding the vertex of T and will be referred to as a trivial surface.

Theorem. Let d be an n-tuple of quadrilateral disk types described above. If [y\vec] is a nonzero nonnegative integer solution to S_d then there exists a unique normal surface F in M with no trivial components such that [x\vec]_F = [y\vec].

This theorem is an extension of the main result of for the case of regular triangulations to the case of singular ones. The proof is actually the same. Note that S_d has n variables and n+1 equations while the number of equations in the usual Haken system is much bigger than the number of variables.

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S.V. Matveev, Additivity of complexity and Haken's method in 3-manifolds topology, Ukraine J. of Math., Vol 41, No 9, 1989, 1234-1239, (Russian)

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J. Tollefson, Normal surface Q-theory, Pacific J. of Math., Vol 183, No 2, 1998, 359-374.

Date received: June 2, 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 # cacy-24.