Unknotting tunnels and Seifert surfaces. (English) Zbl 1047.57008

Scharlemann, Martin; Thompson, Abigail

Proc. Lond. Math. Soc., III. Ser. 87, No. 2, 523-544 (2003).

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 \(\rho = 1\).

Reviewer: Vincent Blanloeil (Strasbourg)

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