Unknotting tunnels and Seifert surfaces
| dc.creator | Scharlemann, Martin | |
| dc.creator | Thompson, Abigail | |
| dc.date | 2000-10-22 | |
| dc.date.accessioned | 2026-07-07T04:38:10Z | |
| dc.date.available | 2026-07-07T04:38:10Z | |
| dc.description | Let $K$ be a knot with an unknotting tunnel $γ$ and suppose that $K$ is not a 2-bridge knot. There is an invariant $ρ= p/q \in \mathbb{Q}/2 \mathbb{Z}$, $p$ odd, defined for the pair $(K, γ)$. The invariant $ρ$ has interesting geometric properties: It is often straightforward to calculate; e. g. for $K$ a torus knot and $γ$ an annulus-spanning arc, $ρ(K, γ) = 1$. Although $ρ$ is defined abstractly, it is naturally revealed when $K \cup γ$ is put in thin position. If $ρ\neq 1$ then there is a minimal genus Seifert surface $F$ for $K$ such that the tunnel $γ$ can be slid and isotoped to lie on $F$. One consequence: if $ρ(K, γ) \neq 1$ then $genus(K) > 1$. This confirms a conjecture of Goda and Teragaito for pairs $(K, γ)$ with $ρ(K, γ) \neq 1$. | |
| dc.description | 29 pages, 20 figures | |
| dc.identifier | https://arxiv.org/abs/math/0010212 | |
| dc.identifier | http://arxiv.org/abs/math/0010212 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/60178 | |
| dc.subject | Geometric Topology | |
| dc.subject | 57M25; 57M27 | |
| dc.title | Unknotting tunnels and Seifert surfaces | |
| dc.type | text |