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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Levels of knotting of spatial handlebodies


Authors: R. Benedetti and R. Frigerio
Journal: Trans. Amer. Math. Soc. 365 (2013), 2099-2167
MSC (2010): Primary 57M27; Secondary 57M15, 57M05
Published electronically: August 21, 2012
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Abstract: If $ H$ is a spatial handlebody, i.e. a handlebody embedded in the $ 3$-sphere, a spine of $ H$ is a graph $ \Gamma \subset S^3$ such that $ H$ is a regular neighbourhood of $ \Gamma $. Usually, $ H$ is said to be unknotted if it admits a planar spine. This suggests that a handlebody should be considered not very knotted if it admits spines that enjoy suitable special properties. Building on this remark, we define several levels of knotting of spatial handlebodies, and we provide a complete description of the relationships between these levels, focusing our attention on the case of genus 2. We also relate the knotting level of a spatial handlebody $ H$ to classical topological properties of its complement $ M=S^3\setminus H$, such as its cut number. More precisely, we show that if $ H$ is not highly knotted, then $ M$ admits special cut systems for $ M$, and we discuss the extent to which the converse implication holds. Along the way we construct obstructions that allow us to determine the knotting level of several families of spatial handlebodies. These obstructions are based on recent quandle-coloring invariants for spatial handlebodies, on the extension to the context of spatial handlebodies of tools coming from the theory of homology boundary links, on the analysis of appropriate coverings of handlebody complements, and on the study of the classical Alexander elementary ideals of their fundamental groups.


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Additional Information

R. Benedetti
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy
Email: benedett@dm.unipi.it

R. Frigerio
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy
Email: frigerio@dm.unipi.it

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05707-2
PII: S 0002-9947(2012)05707-2
Keywords: Quandle, maximal free covering, Alexander polynomial, cut number, corank, spatial graph, boundary link, homology boundary link, pattern
Received by editor(s): March 21, 2011
Received by editor(s) in revised form: September 2, 2011
Published electronically: August 21, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.