Download e-book for kindle: Algorithmic Topology and Classification of 3-Manifolds by Sergei Matveev
By Sergei Matveev
From the reports of the first version: "This booklet presents a complete and particular account of alternative themes in algorithmic third-dimensional topology, culminating with the popularity technique for Haken manifolds and together with the up to date ends up in machine enumeration of 3-manifolds. Originating from lecture notes of varied classes given via the writer over a decade, the ebook is meant to mix the pedagogical technique of a graduate textbook (without routines) with the completeness and reliability of a study monograph… all of the fabric, with few exceptions, is gifted from the ordinary perspective of specific polyhedra and certain spines of 3-manifolds. This selection contributes to maintain the extent of the exposition particularly straight forward. In end, the reviewer subscribes to the citation from the again disguise: "the booklet fills a spot within the latest literature and should develop into a regular reference for algorithmic third-dimensional topology either for graduate scholars and researchers". Zentralblatt f?r Mathematik 2004 For this second variation, new effects, new proofs, and commentaries for a greater orientation of the reader were additional. specifically, in bankruptcy 7 numerous new sections pertaining to purposes of the pc software "3-Manifold Recognizer" were integrated.
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Extra info for Algorithmic Topology and Classification of 3-Manifolds (Algorithms and Computation in Mathematics)
Then we may assume that they coincide. 26. Suppose that Pa , Pb are obtained by two of them. Then one can transform Pa into Pb by a composition of moves β −1 , α, T −1 , and β as shown in Fig. 41. 28 to split β ±1 and α into compositions of moves T ±1 , U . If a and b lie in edges of P having a common vertex, the transformation of Pa to Pb is carried out by β −1 and β: we create a U -turn instead of the loop at a, and then replace it by a loop at b. It follows that we can move loops along the singular graph SP wherever we like.
The normal bundle of the boundary curve of the smaller 2-component c is trivial while the one of c is nontrivial. 11, we apply moves T ±1 to enlarge c and diminish c until we get a loop, see Fig. 43. 25. Let P, Q be special polyhedra such that P ∼Q and Q is unthickenable. 31, we may assume that Q has a loop. 8, one can ﬁnd a sequence of moves T ±1 , U ±1 transforming P into Q. We replace each move U −1 that occurs in the sequence by the move β, 46 1 Simple and Special Polyhedra Fig. 43. 28 is a composition of moves T ±1 , U .
The last move in the Fig. 8 is a composition of T, T −1 . 44 1 Simple and Special Polyhedra Fig. 39. A substitute for U −1 : the move β Fig. 40. 29 is that loops are very movable: One may transfer them from one place to another using T ±1 and U , but not U −1 . 29. Let a, b be two triple points of a special polyhedron P . Suppose that special polyhedra Pa and Pb are obtained from P by creating loops at a and b, respectively. Then one can transform Pa into Pb by moves T ±1 , U . Proof. We ﬁrst consider the case when a and b lie in the same edge of P .
Algorithmic Topology and Classification of 3-Manifolds (Algorithms and Computation in Mathematics) by Sergei Matveev