Minimal crystallizations of 3-manifolds

Basak, Biplab and Datta, Basudeb (2014) Minimal crystallizations of 3-manifolds. In: ELECTRONIC JOURNAL OF COMBINATORICS, 21 (1).

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Abstract

We have introduced the weight of a group which has a presentation with number of relations is at most the number of generators. We have shown that the number of facets of any contracted pseudotriangulation of a connected closed 3-manifold M is at least the weight of the fundamental group of M. This lower bound is sharp for the 3-manifolds RP3, L(3, 1), L(5, 2), S-1 x S-1 x S-1, S-2 x S-1, S-2 (x) under bar S-1 and S-3/Q(8), where Q(8) is the quaternion group. Moreover, there is a unique such facet minimal pseudotriangulation in each of these seven cases. We have also constructed contracted pseudotriangulations of L(kq - 1, q) with 4(q + k - 1) facets for q >= 3, k >= 2 and L(kq + 1, q) with 4(q + k) facets for q >= 4, k >= 1. By a recent result of Swartz, our pseudotriangulations of L(kg + 1, q) are facet minimal when kg + 1 are even. In 1979, Gagliardi found presentations of the fundamental group of a manifold M in terms of a contracted pseudotriangulation of M. Our construction is the converse of this, namely, given a presentation of the fundamental group of a 3-manifold M, we construct a contracted pseudotriangulation of M. So, our construction of a contracted pseudotriangulation of a 3-manifold M is based on a presentation of the fundamental group of M and it is computer-free.

Item Type:	Journal Article
Publication:	ELECTRONIC JOURNAL OF COMBINATORICS
Publisher:	ELECTRONIC JOURNAL OF COMBINATORICS
Additional Information:	Copyright for this article belongs to the ELECTRONIC JOURNAL OF COMBINATORICS, USA
Keywords:	Pseudotriangulations of manifolds; Crystallizations; Lens spaces; Presentations of groups
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	26 May 2014 11:25
Last Modified:	26 May 2014 11:25
URI:	http://eprints.iisc.ac.in/id/eprint/48977

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