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

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

2011-06-24

arxiv.org/abs/1106.4952v1

Mathematics

Geometric Topology

15 pages, 6 figures

Scientific paper

A theory of complexity for pairs (M,G) with M an arbitrary closed 3-manifold and G a 3-valent graph in M 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 a knot in M, we also prove that any function that is fully additive under connected sum along knots is actually a function of the ambient manifold only.

Affiliated with

Also associated with

No associations

Rating

Notes on the complexity of 3-valent graphs in 3-manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Notes on the complexity of 3-valent graphs in 3-manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Notes on the complexity of 3-valent graphs in 3-manifolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-394026