Unknotting tunnels and Seifert surfaces

Details Unknotting tunnels and Seifert surfaces Unknotting tunnels and Seifert surfaces

2000-10-22

arxiv.org/abs/math/0010212v1

Mathematics

Geometric Topology

29 pages, 20 figures

Scientific paper

Let $K$ be a knot with an unknotting tunnel $\gamma$ and suppose that $K$ is not a 2-bridge knot. There is an invariant $\rho = p/q \in \mathbb{Q}/2 \mathbb{Z}$, $p$ odd, defined for the pair $(K, \gamma)$. The invariant $\rho$ has interesting geometric properties: It is often straightforward to calculate; e. g. for $K$ a torus knot and $\gamma$ an annulus-spanning arc, $\rho(K, \gamma) = 1$. Although $\rho$ is defined abstractly, it is naturally revealed when $K \cup \gamma$ is put in thin position. If $\rho \neq 1$ then there is a minimal genus Seifert surface $F$ for $K$ such that the tunnel $\gamma$ can be slid and isotoped to lie on $F$. One consequence: if $\rho(K, \gamma) \neq 1$ then $genus(K) > 1$. This confirms a conjecture of Goda and Teragaito for pairs $(K, \gamma)$ with $\rho(K, \gamma) \neq 1$.

Affiliated with

Also associated with

No associations

Rating

Unknotting tunnels and Seifert surfaces 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 Unknotting tunnels and Seifert surfaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Unknotting tunnels and Seifert surfaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-46084