Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1706.02143

Minimal 4-colored graphs representing an infinite family of hyperbolic 3-manifolds

P. Cristofori·E. Fominykh·M. Mulazzani·V. Tarkaev
View original source ↗Read source document ↗Export metadata ↓

Abstract

The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity 32) and for compact orientable 3-manifolds with toric boundary (up to graph complexity 12) and for infinite families of lens spaces. In this paper we extend to graph complexity 14 the computations for orientable manifolds with toric boundary and we give two-sided bounds for the graph complexity of tetrahedral manifolds. As a consequence, we compute the exact value of this invariant for an infinite family of such manifolds.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

P. Cristofori, E. Fominykh, M. Mulazzani, V. Tarkaev. 2017-12-05. Minimal 4-colored graphs representing an infinite family of hyperbolic 3-manifolds. https://doi.org/10.1007/s13398-017-0478-4

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Cut pairs and Morse splitting of finitely generated groups

Bowditch's theorem for hyperbolic groups establishes a fundamental correspondence between splittings over two-ended subgroups and the existence of local cut points in the Gromov boundary. While analogous results have been obtained for CAT(0) and relatively hyperbolic groups, no general theorem of this type exists for arbitrary finitely generated groups. The Morse boundary, introduced by Charney-Sultan and extended by Cordes, provides a quasi-isometry invariant boundary for any finitely generated group that naturally generalizes the Gromov boundary. In this paper, we prove that a splitting of a finitely generated group with connected Morse boundary over a two-ended Morse subgroup gives rise to a separating pair of points in the Morse boundary.

math.GT↗
arXiv2026

Khovanov Homology in Connected Sums

Khovanov homology is an invariant for links in the three sphere that categorizes the Jones polynomial. We extend Khovanov's construction to links in 3-manifolds that are connected sums of orientable interval bundles over surfaces. Cutting the 3-manifold along a separating sphere, we construct type D and type A structures that are invariants of tangles in the two halves following the work of Roberts. Gluing the type D and type A structures along the common boundary recovers the Khovanov homology of the link.

math.GT↗
arXiv2026

Fox-Milnor condition for concordant knots in homology 3-spheres

This paper will show that the Alexander polynomial of a knot, which is of slice type in an oriented homology 3-sphere, obeys the Fox-Milnor polynomial condition. A relation between Alexander polynomial of concordant knots in an oriented homology 3-sphere is established.

math.GT↗