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Knots and \(k\)-width. (English) Zbl 1189.57005

For a generic closed curve \(\gamma\) in \(\mathbb{R}^3\), the bridge number \(b(\gamma)\) is defined to be the number of maxima with respect to the \(z\)-axis, and the bridge number of a knot in \(\mathbb{R}^3\) is the minimum of the bridge numbers of closed curves that represent the knot.
Let \(\mathcal{S}_1\) be the set of all flat planes in \(\mathbb{R}^3\) that are parallel to the \(xy\)-plane. Then \(\mathcal{S}_1\) is identified with a line. Given a generic closed curve \(\gamma\), let \(\mathcal{T}_1(\gamma)\subset\mathcal{S}_1\) be the set of planes transverse to \(\gamma\). It forms an open subset in \(\mathcal{S}_1\). If we define the \(1\)-bridge number \(b_1(\gamma)\) to be the number of connected components of \(\mathcal{T}_1(\gamma)\), then \(b_1(\gamma)=2b(\gamma)+1\).
Similarly, let \(\mathcal{S}_2\) be the set of all flat planes in \(\mathbb{R}^3\) that are parallel to the \(z\)-axis, \(\mathcal{S}_3\) the set of all flat planes in \(\mathbb{R}^3\), and \(\mathcal{S}_4\) the set of all flat planes and round two-spheres in \(\mathbb{R}^3\). For a generic (in an appropriate sense) closed curve \(\gamma\), let \(\mathcal{T}_k(\gamma)\) be the open set in \(\mathcal{S}_k\) that consists of planes or spheres transverse to \(\gamma\).
The authors define the \(k\)-bridge number \(b_k(\gamma)\) to be the number of connected components of \(\mathcal{T}_k(\gamma)\). For a knot, the \(k\)-bridge number is defined similarly.
The authors also generalize the width of a knot using \(\mathcal{S}_k\). For a generic closed curve \(\gamma\), define \(w_k(\gamma):=\sum_{P_i\in\mathcal{T}_k(\gamma)}\sharp(P_i\cap\gamma)\). Note that \(w_1(\gamma)\) is the width introduced by D. Gabai [J. Differ. Geom. 26, 479–536 (1987; Zbl 0639.57008)].
In the paper under review the authors study \(k\)-bridge number and \(k\)-width.
They show that for a given integer \(n\), the number of knots with \(2\)-bridge number \(n\) or less is finite. They also show that the \(2\)-width has the same property. In particular, it is shown that if a knot has \(2\)-bridge number less than \(7\) or \(2\)-width less than \(11\), then it is either the unknot or the trefoil.
Lower bounds for \(b_2\) and \(w_2\) are given in terms of the total curvature of the plane curve projected to the \(xy\)-plane, contrary to the bridge number; one can construct a closed curve with a fixed bridge number with arbitrarily large total curvature.
Some observations for \(b_3\), \(w_3\), \(b_4\) and \(w_4\) are also given.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)

Citations:

Zbl 0639.57008
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References:

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