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Unknotting tunnels and Seifert surfaces. (English) Zbl 1047.57008
The authors consider knots $$K$$ with an unknotting tunnel $$\gamma$$. They investigate deeper a conjecture of Morimoto in which he stated that if a knot $$K\subset S^3$$ has a single unknotting tunnel $$\gamma$$, then $$\gamma$$ can be moved to be level with respect to the natural height function on $$K$$ given by a minimal presentation of $$K$$. The proof of this conjecture is based on a “thinning” process of the $$1$$-complex $$K\cup\gamma$$ which can simplify the presentation of this $$1$$-complex until the tunnel can be moved either to a level arc or a level circuit. In this paper the authors construct an obstruction $$\rho\in {\mathbb Q}_{\displaystyle /2{\mathbb Z}}$$ to further useful motion of $$\gamma$$ in the “thinning” process. When the knot is not a $$2$$-bridge knot the obstruction $$\rho$$ can be defined in a way independent of the thin position, thereby $$\rho$$ can be viewed as an invariant of the pair $$(K,\gamma)$$. This new invariant is easy to compute. Moreover when $$\rho\not = 1$$ the tunnel can be isotoped onto a minimal genus Seifert surface, and then allows to prove a conjecture of H. Goda and M. Teragaito. The authors announce that in a forthcoming paper in Trans. Am. Math. Soc. they will focus on the case $$\rho = 1$$.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)
##### Keywords:
knots; tunnel; invariant of knots; Seifert surface
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