A theory of complexity for pairs (M,G) with M an arbitrary closed 3-manifold and G in M a 3-valent graph was introduced by the first two named authors, extending the original notion due to Matveev. The complexity c is known to be always additive under connected sum away from the graphs, but not always under connected sum along (unknotted) arcs of the graphs. In this article, we prove the slightly surprising fact that if in M there is a sphere intersecting G transversely at one point, and this point belongs to an edge e of G, then e can be canceled from G without affecting the complexity. Using this fact we completely characterize the circumstances under which complexity is additive under connected sum along graphs. For the set of pairs (M,K) with K in M a knot, we also prove that any function that is fully additive under connected sum along knots is actually a function of the ambient manifold only.

Notes on the complexity of 3-valent graphs in 3-manifolds

PERVOVA, EKATERINA;PETRONIO, CARLO;
2012-01-01

Abstract

A theory of complexity for pairs (M,G) with M an arbitrary closed 3-manifold and G in M a 3-valent graph was introduced by the first two named authors, extending the original notion due to Matveev. The complexity c is known to be always additive under connected sum away from the graphs, but not always under connected sum along (unknotted) arcs of the graphs. In this article, we prove the slightly surprising fact that if in M there is a sphere intersecting G transversely at one point, and this point belongs to an edge e of G, then e can be canceled from G without affecting the complexity. Using this fact we completely characterize the circumstances under which complexity is additive under connected sum along graphs. For the set of pairs (M,K) with K in M a knot, we also prove that any function that is fully additive under connected sum along knots is actually a function of the ambient manifold only.
2012
Pervova, Ekaterina; Petronio, Carlo; Sasso, V.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/190453
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact