Mathematics – Geometric Topology
Scientific paper
2009-03-26
Mathematics
Geometric Topology
19pages, 7figures
Scientific paper
We consider non-orientable closed surfaces of minimum crosscap number in the $(p,q)$-lens space $L(p,q) \cong V_1 \cup_{\partial} V_2$, where $V_1$ and $V_2$ are solid tori. Bredon and Wood gave a formula for calculating the minimum crosscap number. Rubinstein showed that $L(p,q)$ with $p$ even has only one isotopy class of such surfaces, and it is represented by a surface in a standard form, which is constructed from a meridian disk in $V_1$ by performing a finite number of band sum operations in $V_1$ and capping off the resulting boundary circle by a meridian disk of $V_2$. We show that the standard form corresponds to an edge-path $\lambda$ in a certain tree graph in the closure of the hyperbolic upper half plane. Let $0=p_0/q_0, p_1/q_1, ..., p_k/q_k = p/q$ be the labels of vertices which $\lambda$ passes. Then the slope of the boundary circle of the surface right after the $i$-th band sum is $(p_i, q_i)$. The number of edges of $\lambda$ is equal to the minimum crosscap number. We give an easy way of calculating $p_i / q_i$ using a certain continued fraction expansion of $p/q$.
Iwakura Miwa
No associations
LandOfFree
Geometrically incompressible non-orientable closed surfaces in lens spaces 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 Geometrically incompressible non-orientable closed surfaces in lens spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Geometrically incompressible non-orientable closed surfaces in lens spaces will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-64990