{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,10]],"date-time":"2025-12-10T12:25:40Z","timestamp":1765369540113,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 $\\mathbb{RP}^3$, $L(3,1)$, $L(5,2)$, $S^1\\times S^1 \\times S^1$, $S^2 \\times S^1$, twisted product of $S^2$\u00a0and $S^1$ 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 $q \\geq 3$, $k \\geq 2$ and $L(kq+1,q)$ with $4(q+k)$ facets for $q\\geq 4$, $k\\geq 1$. By a recent result of Swartz, our pseudotriangulations of $L(kq+1, q)$ are facet minimal when $kq+1$ 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.<\/jats:p>","DOI":"10.37236\/3956","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:06:59Z","timestamp":1578704819000},"source":"Crossref","is-referenced-by-count":10,"title":["Minimal Crystallizations of 3-Manifolds"],"prefix":"10.37236","volume":"21","author":[{"given":"Biplab","family":"Basak","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Basudeb","family":"Datta","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2014,3,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i1p61\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i1p61\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:01:42Z","timestamp":1579258902000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v21i1p61"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,3,17]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2014,1,13]]}},"URL":"https:\/\/doi.org\/10.37236\/3956","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2014,3,17]]},"article-number":"P1.61"}}