## 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)

Zbl 0639.57008
Full Text:

### References:

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