Thinning genus two Heegaard spines in the 3-sphere
We study trivalent graphs in $S^{3}$ whose closed complement is a genus two handlebody. We show that such a graph, when put in thin position, has a simple (i. e. non-loop) level edge.
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arXiv subjects
Abigail Thompson
Publications and source records attributed to Abigail Thompson.
Unknotting tunnels and Seifert surfaces
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 $ρ(K, γ) \neq 1$.
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Levelling an unknotting tunnel
It is a consequence of theorems of Gordon-Reid [Tangle decompositions of tunnel number one knots and links, J. Knot Theory and its Ramifications, 4 (1995) 389-409] and Thompson [Thin position and bridge number for knots in the 3-sphere, Topology, 36 (1997) 505-507] that a tunnel number one knot, if put in thin position, will also be in bridge position. We show that in such a thin presentation, the tunnel can be made level so that it lies in a level sphere. This settles a question raised by Morimoto [A note on unknotting tunnels for 2-bridge knots, Bulletin of Faculty of Engineering Takushoku University, 3 (1992) 219-225], who showed that the (now known) classification of unknotting tunnels for 2-bridge knots would follow quickly if it were known that any unknotting tunnel can be made level.
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