No Access

COMPLEXITY OF LINKS IN 3-MANIFOLDS

EKATERINA PERVOVA

Dipartimento di Matematica Applicata, Università di Pisa, Via Filippo Buonarroti, 1C, 56127 Pisa, Italy

Search for more papers by this author

and

CARLO PETRONIO

Dipartimento di Matematica Applicata, Università di Pisa, Via Filippo Buonarroti, 1C, 56127 Pisa, Italy

Search for more papers by this author

https://doi.org/10.1142/S0218216509007531Cited by:3 (Source: Crossref)

Abstract

We introduce a complexity c(X) ∈ ℕ for pairs X = (M,L), where M is a closed orientable 3-manifold and L ⊂ M is a link. The definition employs simple spines, but for well-behaved X's, we show that c(X) equals the minimal number of tetrahedra in a triangulation of M containing L in its 1-skeleton. Slightly adapting Matveev's recent theory of roots for graphs, we carefully analyze the behaviour of c under connected sum away from and along the link. We show in particular that c is almost always additive, describing in detail the circumstances under which it is not. To do so we introduce a certain (0, 2)-root for a pair X, we show that it is well-defined, and we prove that X has the same complexity as its (0, 2)-root. We then consider, for links in the 3-sphere, the relations of c with the crossing number and with the hyperbolic volume of the exterior, establishing various upper and lower bounds. We also specialize our analysis to certain infinite families of links, providing rather accurate asymptotic estimates.

Dedicated to José Maria Montesinos on the occasion of his 65th birthday

Keywords:

AMSC: Primary 57M27, Secondary 57M25, Secondary 57M20

References

S. Anisov, Complexity of torus bundles over the circle with monodromy $(2\quad 1\choose 1\quad 1)^2$, preprint , arXiv:math.GT/0203215 . Google Scholar
R. Benedetti and C. Petronio , Lectures on Hyperbolic Geometry , Universitext ( Springer-Verlag , Berlin, Heidelberg, New York , 1992 ) . Crossref, Google Scholar
G. Burde and H. Zieschang , Knots , Studies in Mathematics 5 ( de Gruyter , Berlin, New York , 1985 ) . Google Scholar
D. Heard, C. Hodgson, B. Martelli and C. Petronio, Hyperbolic graphs of small complexity, preprint, , to appear in Experiment Math , arXiv:0804.4790 . Google Scholar
H. Helling, A. C. Kim and J. L. Mennicke, J. Lie Theory 8, 1 (1999). Google Scholar
C. Hog-Angeloni and S. Matveev, The Zieschang Gedenkschrift, Geometry and Topology Monographs 14 (Geometry and Topology Publications, Coventry, 2008) pp. 295–319. Crossref, Google Scholar
P. K. Kim and J. L. Tollefson, Michigan Math. J. 27, 259 (1980). Web of Science, Google Scholar
M. Lackenby, Proc. London Math. Soc. 88, 204 (2004), DOI: 10.1112/S0024611503014291. Crossref, Web of Science, Google Scholar
W. B. R. Lickorish , An Introduction to Knot Theory , Graduate Texts in Mathematics 175 ( Springer-Verlag , Berlin, Heidelberg, New York , 1997 ) . Crossref, Google Scholar
S. Matveev, Acta Appl. Math. 19, 101 (1990). Crossref, Web of Science, Google Scholar
S. Matveev, Roots of knotted graphs and orbifolds, preprint, MPI 51-2005 . Google Scholar
S. Matveev and E. Pervova, Dokl. Akad. Nauk 378, 151 (2001). Google Scholar
S. Matveev, C. Petronio and A. Vesnin, J. Austral. Math. Soc. 86, 205 (2009), DOI: 10.1017/S1446788708000499. Crossref, Web of Science, Google Scholar
A. Mednykh and A. Vesnin, Hyperbolic 3-manifolds as 2-fold coverings according to Montesinos, preprint 95–010, Universität Bielefeld, Bielefeld, 1995 . Google Scholar
C. Petronio, Topol. Appl. 153, 1658 (2006), DOI: 10.1016/j.topol.2005.06.002. Crossref, Web of Science, Google Scholar
C. Petronio, Math. Proc. Cambridge Philos. Soc. 142, 269 (2007), DOI: 10.1017/S0305004106009807. Crossref, Web of Science, Google Scholar
C. P. Rourke and B. J. Sanderson , Introduction to Piecewise-Linear Topology , Ergebnisse der Mathematik und ihrer Grenzgebiete 69 ( Springer-Verlag , New York, Heidelberg , 1972 ) . Crossref, Google Scholar
W. P. Thurston , The Geometry and Topology of 3-Manifolds ( Princeton , 1978 ) , http://www.msri.org/publications/books/gt3m/ . Google Scholar
R. Williams, Braids (Santa Cruz, CA, 1986) (American Mathematical Society, Providence, RI, 1988) pp. 697–703. Google Scholar